Editing Telescoping series (section)

== Related concepts==
A ''telescoping product'' is a finite [[Product (mathematics)|product]] (or the partial product of an infinite product) that can be canceled by the method of quotients to be eventually only a finite number of factors.<ref name="Brilliant">{{cite web |title=Telescoping Series - Product |url=https://brilliant.org/wiki/telescoping-series-product/ |website=Brilliant Math & Science Wiki |publisher=Brilliant.org |access-date=9 February 2020 |language=en-us}}</ref><ref>{{cite web |last1=Bogomolny |first1=Alexander |title=Telescoping Sums, Series and Products |url=https://www.cut-the-knot.org/m/Algebra/TelescopingSums.shtml |website=Cut the Knot |access-date=9 February 2020}}</ref> It is the finite products in which consecutive terms cancel denominator with numerator, leaving only the initial and final terms. Let <math>a_n</math> be a sequence of numbers. Then, 
<math display="block"> \prod_{n=1}^N \frac{a_{n-1}}{a_n} =  \frac{a_0}{a_N}.</math>
If <math>a_n </math> converges to 1, the resulting product gives:
<math display="block"> \prod_{n=1}^\infty \frac{a_{n-1}}{a_n} = a_0</math>

For example, the infinite product<ref name="Brilliant"/>
<math display="block">\prod_{n=2}^{\infty} \left(1-\frac{1}{n^2} \right)</math>
simplifies as
<math display="block">\begin{align}
\prod_{n=2}^{\infty} \left(1-\frac{1}{n^2} \right)
&=\prod_{n=2}^{\infty}\frac{(n-1)(n+1)}{n^2}
\\
&=\lim_{N\to\infty} \prod_{n=2}^{N}\frac{n-1}{n} \times \prod_{n=2}^{N}\frac{n+1}{n}
\\
&= \lim_{N\to\infty} \left\lbrack {\frac{1}{2} \times \frac{2}{3}  \times \frac{3}{4} \times \cdots \times \frac{N-1}{N}} \right\rbrack
\times \left\lbrack {\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{N}{N-1} \times \frac{N+1}{N}} \right\rbrack
\\
&= \lim_{N\to\infty} \left\lbrack \frac{1}{2} \right\rbrack \times \left\lbrack \frac{N+1}{N} \right\rbrack
\\
&= \frac{1}{2}\times \lim_{N\to\infty} \left\lbrack \frac{N+1}{N} \right\rbrack
\\
&=\frac{1}{2}.
\end{align}</math>