Binomial theorem

Template:Short description Template:CS1 config Template:Image frame In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power Template:Tmath expands into a polynomial with terms of the form Template:Tmath, where the exponents Template:Tmath and Template:Tmath are nonnegative integers satisfying Template:Tmath and the coefficient Template:Tmath of each term is a specific positive integer depending on Template:Tmath and Template:Tmath. For example, for Template:Tmath, <math display=block>(x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. </math>

The coefficient Template:Tmath in each term Template:Tmath is known as the binomial coefficient Template:Tmath or Template:Tmath (the two have the same value). These coefficients for varying Template:Tmath and Template:Tmath can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where Template:Tmath gives the number of different combinations (i.e. subsets) of Template:Tmath elements that can be chosen from an Template:Tmath-element set. Therefore Template:Tmath is usually pronounced as "Template:Tmath choose Template:Tmath".

Statement

According to the theorem, the expansion of any nonnegative integer power Template:Mvar of the binomial Template:Math is a sum of the form <math display="block">(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1} y^1 + {n \choose 2}x^{n-2} y^2 + \cdots + {n \choose n}x^0 y^n,</math> where each <math> \tbinom nk </math> is a positive integer known as a binomial coefficient, defined as

<math display=block>\binom nk = \frac{n!}{k!\,(n-k)!} = \frac{n(n-1)(n-2)\cdots(n-k + 1)}{k(k-1)(k-2)\cdots2\cdot1}.</math>

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as <math display="block">(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}.</math>

The final expression follows from the previous one by the symmetry of Template:Mvar and Template:Mvar in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, <math display=inline>\binom nk = \binom n{n-k}.</math>

A simple variant of the binomial formula is obtained by substituting Template:Math for Template:Mvar, so that it involves only a single variable. In this form, the formula reads <math display=block>\begin{align} (x+1)^n &= {n \choose 0}x^0 + {n \choose 1}x^1 + {n \choose 2}x^2 + \cdots + {n \choose n}x^n \\[4mu] &= \sum_{k=0}^n {n \choose k}x^k. \vphantom{\Bigg)} \end{align}</math>

Examples

The first few cases of the binomial theorem are: <math display="block">\begin{align} (x+y)^0 & = 1, \\[8pt] (x+y)^1 & = x + y, \\[8pt] (x+y)^2 & = x^2 + 2xy + y^2, \\[8pt] (x+y)^3 & = x^3 + 3x^2y + 3xy^2 + y^3, \\[8pt] (x+y)^4 & = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4, \end{align}</math> In general, for the expansion of Template:Math on the right side in the Template:Mvarth row (numbered so that the top row is the 0th row):

• the exponents of Template:Mvar in the terms are Template:Math (the last term implicitly contains Template:Math);
• the exponents of Template:Mvar in the terms are Template:Math (the first term implicitly contains Template:Math);
• the coefficients form the Template:Mvarth row of Pascal's triangle;
• before combining like terms, there are Template:Math terms Template:Math in the expansion (not shown);
• after combining like terms, there are Template:Math terms, and their coefficients sum to Template:Math.

An example illustrating the last two points: <math display="block">\begin{align} (x+y)^3 & = xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy & (2^3 \text{ terms}) \\

       & = x^3 + 3x^2y + 3xy^2 + y^3 & (3 + 1 \text{ terms})

\end{align}</math> with <math>1 + 3 + 3 + 1 = 2^3</math>.

A simple example with a specific positive value of Template:Math: <math display="block">\begin{align} (x+2)^3 &= x^3 + 3x^2(2) + 3x(2)^2 + 2^3 \\ &= x^3 + 6x^2 + 12x + 8. \end{align}</math>

A simple example with a specific negative value of Template:Math: <math display="block">\begin{align} (x-2)^3 &= x^3 - 3x^2(2) + 3x(2)^2 - 2^3 \\ &= x^3 - 6x^2 + 12x - 8. \end{align}</math>

Geometric explanation

File:Binomial theorem visualisation.svg

Visualisation of binomial expansion up to the 4th power

For positive values of Template:Mvar and Template:Mvar, the binomial theorem with Template:Math is the geometrically evident fact that a square of side Template:Math can be cut into a square of side Template:Mvar, a square of side Template:Mvar, and two rectangles with sides Template:Mvar and Template:Mvar. With Template:Math, the theorem states that a cube of side Template:Math can be cut into a cube of side Template:Mvar, a cube of side Template:Mvar, three Template:Math rectangular boxes, and three Template:Math rectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative <math>(x^n)'=nx^{n-1}:</math><ref name="barth2004">Template:Cite journal</ref> if one sets <math>a=x</math> and <math>b=\Delta x,</math> interpreting Template:Mvar as an infinitesimal change in Template:Mvar, then this picture shows the infinitesimal change in the volume of an Template:Mvar-dimensional hypercube, <math>(x+\Delta x)^n,</math> where the coefficient of the linear term (in <math>\Delta x</math>) is <math>nx^{n-1},</math> the area of the Template:Mvar faces, each of dimension Template:Math: <math display="block">(x+\Delta x)^n = x^n + nx^{n-1}\Delta x + \binom{n}{2}x^{n-2}(\Delta x)^2 + \cdots.</math> Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, <math>(\Delta x)^2</math> and higher, become negligible, and yields the formula <math>(x^n)'=nx^{n-1},</math> interpreted as

"the infinitesimal rate of change in volume of an Template:Mvar-cube as side length varies is the area of Template:Mvar of its Template:Math-dimensional faces".

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral <math>\textstyle{\int x^{n-1}\,dx = \tfrac{1}{n} x^n}</math> – see proof of Cavalieri's quadrature formula for details.<ref name="barth2004" />

Binomial coefficients

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written <math>\tbinom{n}{k},</math> and pronounced "Template:Mvar choose Template:Mvar".

Formulas

The coefficient of Template:Math is given by the formula <math display="block">\binom{n}{k} = \frac{n!}{k! \; (n-k)!},</math> which is defined in terms of the factorial function Template:Math. Equivalently, this formula can be written <math display="block">\binom{n}{k} = \frac{n (n-1) \cdots (n-k+1)}{k (k-1) \cdots 1} = \prod_{\ell=1}^k \frac{n-\ell+1}{\ell} = \prod_{\ell=0}^{k-1} \frac{n-\ell}{k - \ell}</math> with Template:Mvar factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient <math>\tbinom{n}{k}</math> is actually an integer.

Combinatorial interpretation

The binomial coefficient <math> \tbinom nk </math> can be interpreted as the number of ways to choose Template:Mvar elements from an Template:Mvar-element set (a combination). This is related to binomials for the following reason: if we write Template:Math as a product <math display="block">(x+y)(x+y)(x+y)\cdots(x+y),</math> then, according to the distributive law, there will be one term in the expansion for each choice of either Template:Mvar or Template:Mvar from each of the binomials of the product. For example, there will only be one term Template:Math, corresponding to choosing Template:Mvar from each binomial. However, there will be several terms of the form Template:Math, one for each way of choosing exactly two binomials to contribute a Template:Mvar. Therefore, after combining like terms, the coefficient of Template:Math will be equal to the number of ways to choose exactly Template:Math elements from an Template:Mvar-element set.

Proofs

Combinatorial proof

Expanding Template:Math yields the sum of the Template:Math products of the form Template:Math where each Template:Math is Template:Mvar or Template:Mvar. Rearranging factors shows that each product equals Template:Math for some Template:Mvar between Template:Math and Template:Mvar. For a given Template:Mvar, the following are proved equal in succession:

• the number of terms equal to Template:Math in the expansion
• the number of Template:Mvar-character Template:Math strings having Template:Mvar in exactly Template:Mvar positions
• the number of Template:Mvar-element subsets of Template:Math
• <math>\tbinom{n}{k},</math> either by definition, or by a short combinatorial argument if one is defining <math>\tbinom{n}{k}</math> as <math>\tfrac{n!}{k! (n-k)!}.</math>

This proves the binomial theorem.

Example

The coefficient of Template:Math in <math display="block">\begin{align}

  (x+y)^3 &= (x+y)(x+y)(x+y) \\
  &= xxx + xxy + xyx + \underline{xyy} + yxx + \underline{yxy} + \underline{yyx} + yyy \\
  &= x^3 + 3x^2y + \underline{3xy^2} + y^3

\end{align}</math> equals <math>\tbinom{3}{2}=3</math> because there are three Template:Math strings of length 3 with exactly two Template:Mvar's, namely, <math display="block">xyy, \; yxy, \; yyx,</math> corresponding to the three 2-element subsets of Template:Math, namely, <math display="block">\{2,3\},\;\{1,3\},\;\{1,2\}, </math> where each subset specifies the positions of the Template:Mvar in a corresponding string.

Inductive proof

Induction yields another proof of the binomial theorem. When Template:Math, both sides equal Template:Math, since Template:Math and <math>\tbinom{0}{0}=1.</math> Now suppose that the equality holds for a given Template:Mvar; we will prove it for Template:Math. For Template:Math, let Template:Math denote the coefficient of Template:Math in the polynomial Template:Math. By the inductive hypothesis, Template:Math is a polynomial in Template:Mvar and Template:Mvar such that Template:Math is <math>\tbinom{n}{k}</math> if Template:Math, and Template:Mvar otherwise. The identity <math display="block"> (x+y)^{n+1} = x(x+y)^n + y(x+y)^n</math> shows that Template:Math is also a polynomial in Template:Mvar and Template:Mvar, and <math display="block"> [(x+y)^{n+1}]_{j,k} = [(x+y)^n]_{j-1,k} + [(x+y)^n]_{j,k-1},</math> since if Template:Math, then Template:Math and Template:Math. Now, the right hand side is <math display="block"> \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k},</math> by Pascal's identity.<ref>Binomial theorem – inductive proofs Template:Webarchive</ref> On the other hand, if Template:Math, then Template:Math and Template:Math, so we get Template:Math. Thus <math display="block">(x+y)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k} x^{n+1-k} y^k,</math> which is the inductive hypothesis with Template:Math substituted for Template:Mvar and so completes the inductive step.

Generalizations

Newton's generalized binomial theorem

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number Template:Mvar, one can define <math display="block">{r \choose k}=\frac{r(r-1) \cdots (r-k+1)}{k!} =\frac{(r)_k}{k!},</math> where <math>(\cdot)_k</math> is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when Template:Mvar is a nonnegative integer. Then, if Template:Mvar and Template:Mvar are real numbers with Template:Math,<ref name=convergence group=Note>This is to guarantee convergence. Depending on Template:Mvar, the series may also converge sometimes when Template:Math.</ref> and Template:Mvar is any complex number, one has <math display="block">\begin{align}

  (x+y)^r & =\sum_{k=0}^\infty {r \choose k} x^{r-k} y^k \\
  &= x^r + r x^{r-1} y + \frac{r(r-1)}{2!} x^{r-2} y^2 + \frac{r(r-1)(r-2)}{3!} x^{r-3} y^3 + \cdots.
\end{align}</math>

When Template:Mvar is a nonnegative integer, the binomial coefficients for Template:Math are zero, so this equation reduces to the usual binomial theorem, and there are at most Template:Math nonzero terms. For other values of Template:Mvar, the series typically has infinitely many nonzero terms.

For example, Template:Math gives the following series for the square root: <math display="block">\sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \frac{7}{256}x^5 - \cdots.</math>

Taking Template:Math, the generalized binomial series gives the geometric series formula, valid for Template:Math: <math display="block">(1+x)^{-1} = \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots.</math>

More generally, with Template:Math, we have for Template:Math:<ref name=wolfram2>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\frac{1}{(1+x)^s} = \sum_{k=0}^\infty {-s \choose k} x^k = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k x^k.</math>

So, for instance, when Template:Math, <math display="block">\frac{1}{\sqrt{1+x}} = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 - \frac{63}{256}x^5 + \cdots.</math>

Replacing Template:Mvar with Template:Mvar yields: <math display="block">\frac{1}{(1-x)^s} = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k (-x)^k = \sum_{k=0}^\infty {s+k-1 \choose k} x^k.</math>

So, for instance, when Template:Math, we have for Template:Math: <math display="block">\frac{1}{\sqrt{1-x}} = 1 + \frac{1}{2}x + \frac{3}{8}x^2 + \frac{5}{16}x^3 + \frac{35}{128}x^4 + \frac{63}{256}x^5 + \cdots.</math>

Further generalizations

The generalized binomial theorem can be extended to the case where Template:Mvar and Template:Mvar are complex numbers. For this version, one should again assume Template:Math<ref name=convergence group=Note /> and define the powers of Template:Math and Template:Mvar using a holomorphic branch of log defined on an open disk of radius Template:Math centered at Template:Mvar. The generalized binomial theorem is valid also for elements Template:Mvar and Template:Mvar of a Banach algebra as long as Template:Math, and Template:Mvar is invertible, and Template:Math.

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant Template:Mvar, define <math> x^{(0)} = 1 </math> and <math display="block"> x^{(n)} = \prod_{k=1}^{n}[x+(k-1)c]</math> for <math> n > 0.</math> Then<ref name="Sokolowsky">Template:Cite journal</ref> <math display="block"> (a + b)^{(n)} = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^{(k)}.</math> The case Template:Math recovers the usual binomial theorem.

More generally, a sequence <math>\{p_n\}_{n=0}^\infty</math> of polynomials is said to be of binomial type if

• <math> \deg p_n = n </math> for all <math>n</math>,
• <math> p_0(0) = 1 </math>, and
• <math> p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) </math> for all <math>x</math>, <math>y</math>, and <math>n</math>.

An operator <math>Q</math> on the space of polynomials is said to be the basis operator of the sequence <math>\{p_n\}_{n=0}^\infty</math> if <math>Qp_0 = 0</math> and <math> Q p_n = n p_{n-1} </math> for all <math> n \geqslant 1 </math>. A sequence <math>\{p_n\}_{n=0}^\infty</math> is binomial if and only if its basis operator is a Delta operator.<ref>Template:Cite book</ref> Writing <math> E^a </math> for the shift by <math> a </math> operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference <math> I - E^{-c} </math> for <math> c>0 </math>, the ordinary derivative for <math> c=0 </math>, and the forward difference <math> E^{-c} - I </math> for <math> c<0 </math>.

Multinomial theorem

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

<math display="block">(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots +k_m = n} \binom{n}{k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}, </math>

where the summation is taken over all sequences of nonnegative integer indices Template:Math through Template:Math such that the sum of all Template:Math is Template:Mvar. (For each term in the expansion, the exponents must add up to Template:Mvar). The coefficients <math> \tbinom{n}{k_1,\cdots,k_m} </math> are known as multinomial coefficients, and can be computed by the formula <math display="block"> \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}.</math>

Combinatorially, the multinomial coefficient <math>\tbinom{n}{k_1,\cdots,k_m}</math> counts the number of different ways to partition an Template:Mvar-element set into disjoint subsets of sizes Template:Math.

Template:Anchor Multi-binomial theorem

When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to <math display="block"> (x_1+y_1)^{n_1}\dotsm(x_d+y_d)^{n_d} = \sum_{k_1=0}^{n_1}\dotsm\sum_{k_d=0}^{n_d} \binom{n_1}{k_1} x_1^{k_1}y_1^{n_1-k_1} \dotsc \binom{n_d}{k_d} x_d^{k_d}y_d^{n_d-k_d}. </math>

This may be written more concisely, by multi-index notation, as <math display="block"> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} x^\nu y^{\alpha - \nu}.</math>

General Leibniz rule

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The general Leibniz rule gives the Template:Mvarth derivative of a product of two functions in a form similar to that of the binomial theorem:<ref>Template:Cite book</ref> <math display="block">(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x).</math>

Here, the superscript Template:Math indicates the Template:Mvarth derivative of a function, <math>f^{(n)}(x) = \tfrac{d^n}{dx^n}f(x)</math>. If one sets Template:Math and Template:Math, cancelling the common factor of Template:Math from each term gives the ordinary binomial theorem.<ref>Template:Cite book</ref>

History

Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent <math>n=2</math>.<ref name="Coolidge">Template:Cite journal</ref> Greek mathematician Diophantus cubed various binomials, including <math>x-1</math>.<ref name="Coolidge" /> Indian mathematician Aryabhata's method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent <math>n=3</math>.<ref name="Coolidge" />

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting Template:Mvar objects out of Template:Mvar without replacement (combinations), were of interest to ancient Indian mathematicians. The Jain Bhagavati Sutra (c. 300 BC) describes the number of combinations of philosophical categories, senses, or other things, with correct results up through Template:Tmath (probably obtained by listing all possibilities and counting them)<ref name=biggs>Template:Cite journal</ref> and a suggestion that higher combinations could likewise be found.<ref>Template:Cite journal Reprinted as "The Mathematical Achievements of the Jainas" in Template:Cite book</ref> The Chandaḥśāstra by the Indian lyricist Piṅgala (3rd or 2nd century BC) somewhat cryptically describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator Halāyudha his "method of pyramidal expansion" (meru-prastāra) for counting metres is equivalent to Pascal's triangle.<ref>Template:Cite journal Template:Pb Template:Cite journal (Preprint) Template:Pb Survey sources: Template:Pb Template:Cite book Template:Pb Template:Cite book Template:Pb Template:Cite book</ref> Varāhamihira (6th century AD) describes another method for computing combination counts by adding numbers in columns.<ref name=gupta>Template:Cite journal Reprinted in Template:Cite book</ref> By the 9th century at latest Indian mathematicians learned to express this as a product of fractions Template:Tmath, and clear statements of this rule can be found in Śrīdhara's Pāṭīgaṇita (8th–9th century), Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850), and Bhāskara II's Līlāvatī (12th century).Template:RTemplate:R<ref>Template:Cite book</ref>

The Persian mathematician al-Karajī (953–1029) wrote a now-lost book containing the binomial theorem and a table of binomial coefficients, often credited as their first appearance.<ref name=yadegari>Template:Cite journal</ref><ref name=rashed>Template:Cite journal Translated into English by A. F. W. Armstrong in Template:Cite book</ref><ref>Template:Cite encyclopedia</ref><ref> Template:Cite journal Republished in Template:Cite book</ref> An explicit statement of the binomial theorem appears in al-Samawʾal's al-Bāhir (12th century), there credited to al-Karajī.Template:RTemplate:R Al-Samawʾal algebraically expanded the square, cube, and fourth power of a binomial, each in terms of the previous power, and noted that similar proofs could be provided for higher powers, an early form of mathematical induction. He then provided al-Karajī's table of binomial coefficients (Pascal's triangle turned on its side) up to Template:Tmath and a rule for generating them equivalent to the recurrence relation Template:Tmath.Template:R<ref name=Karaji>Template:MacTutor</ref> The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.<ref name="Coolidge" /> The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and also Chu Shih-Chieh.<ref name="Coolidge" /> Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.<ref>Template:Cite book</ref>

In Europe, descriptions of the construction of Pascal's triangle can be found as early as Jordanus de Nemore's De arithmetica (13th century).<ref>Template:Cite journal</ref> In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express <math>(1+x)^n</math> in terms of <math>(1+x)^{n-1}</math>, via "Pascal's triangle".<ref name=Kline>Template:Cite book</ref> Other 16th century mathematicians including Niccolò Fontana Tartaglia and Simon Stevin also knew of it.<ref name=Kline /> 17th-century mathematician Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique.<ref>Template:Cite book</ref>

By the early 17th century, some specific cases of the generalized binomial theorem, such as for <math>n=\tfrac{1}{2}</math>, can be found in the work of Henry Briggs' Arithmetica Logarithmica (1624).Template:R Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of John Wallis's Arithmetic Infinitorum and his method of interpolation.<ref name=Kline /><ref>Template:Cite book</ref><ref name="Coolidge" /><ref>Template:Cite journal</ref>Template:R A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670.<ref name=stillwell>Template:Cite book</ref>

Applications

Multiple-angle identities

For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula, <math display="block">\cos\left(nx\right)+i\sin\left(nx\right) = \left(\cos x+i\sin x\right)^n.</math>

Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for Template:Math and Template:Math. For example, since <math display="block">\left(\cos x + i\sin x\right)^2 = \cos^2 x + 2i \cos x \sin x - \sin^2 x = (\cos^2 x-\sin^2 x) + i(2\cos x\sin x),</math> But De Moivre's formula identifies the left side with <math>(\cos x+i\sin x)^2 = \cos(2x)+i\sin(2x)</math>, so <math display="block">\cos(2x) = \cos^2 x - \sin^2 x \quad\text{and}\quad\sin(2x) = 2 \cos x \sin x,</math> which are the usual double-angle identities. Similarly, since <math display="block">\left(\cos x + i\sin x\right)^3 = \cos^3 x + 3i \cos^2 x \sin x - 3 \cos x \sin^2 x - i \sin^3 x,</math> De Moivre's formula yields <math display="block">\cos(3x) = \cos^3 x - 3 \cos x \sin^2 x \quad\text{and}\quad \sin(3x) = 3\cos^2 x \sin x - \sin^3 x.</math> In general, <math display="block">\cos(nx) = \sum_{k\text{ even}} (-1)^{k/2} {n \choose k}\cos^{n-k} x \sin^k x</math> and <math display="block">\sin(nx) = \sum_{k\text{ odd}} (-1)^{(k-1)/2} {n \choose k}\cos^{n-k} x \sin^k x.</math>There are also similar formulas using Chebyshev polynomials.

Series for e

The [[e (mathematical constant)|number Template:Mvar]] is often defined by the formula <math display="block">e = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n.</math>

Applying the binomial theorem to this expression yields the usual infinite series for Template:Mvar. In particular: <math display="block">\left(1 + \frac{1}{n}\right)^n = 1 + {n \choose 1}\frac{1}{n} + {n \choose 2}\frac{1}{n^2} + {n \choose 3}\frac{1}{n^3} + \cdots + {n \choose n}\frac{1}{n^n}.</math>

The Template:Mvarth term of this sum is <math display="block">{n \choose k}\frac{1}{n^k} = \frac{1}{k!}\cdot\frac{n(n-1)(n-2)\cdots (n-k+1)}{n^k}</math>

As Template:Math, the rational expression on the right approaches Template:Math, and therefore <math display="block">\lim_{n\to\infty} {n \choose k}\frac{1}{n^k} = \frac{1}{k!}.</math>

This indicates that Template:Mvar can be written as a series: <math display="block">e=\sum_{k=0}^\infty\frac{1}{k!}=\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots.</math>

Indeed, since each term of the binomial expansion is an increasing function of Template:Mvar, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to Template:Mvar.

Probability

The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials <math>\{X_t\}_{t\in S}</math> with probability of success <math>p\in [0,1]</math> all not happening is

<math> P\biggl(\bigcap_{t\in S} X_t^C\biggr) = (1-p)^{|S|} = \sum_{n=0}^{|S|} {|S| \choose n} (-p)^n.</math>

An upper bound for this quantity is <math> e^{-p|S|}.</math><ref>Template:Cite book </ref>

In abstract algebra

The binomial theorem is valid more generally for two elements Template:Math and Template:Math in a ring, or even a semiring, provided that Template:Math. For example, it holds for two Template:Math matrices, provided that those matrices commute; this is useful in computing powers of a matrix.<ref>Template:Cite book</ref>

The binomial theorem can be stated by saying that the polynomial sequence Template:Math is of binomial type.

See also

• Binomial approximation
• Binomial distribution
• Binomial inverse theorem
• Binomial coefficient
• Stirling's approximation
• Tannery's theorem
• Polynomials calculating sums of powers of arithmetic progressions
• q-binomial theorem

Further reading

• Template:Cite book

External links

• Template:SpringerEOM
• Binomial Theorem by Stephen Wolfram, and "Binomial Theorem (Step-by-Step)" by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007.
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