Editing Euler's formula (section)

===Using power series===
[[File:Eulers-forrmula-standalone.svg|thumb|alt=Each successive term in the series rotates 90 degrees counter clockwise. The even-power terms are real, hence parallel to the real line, and the odd-power terms are imaginary, hence parallel to the imaginary axis. Plotting each term as a vectors in the complex plane lying end-to-end (vector addition) results in a piecewise-linear spiral starting from the origin and converging to the point (cos 2, sin 2) on the unit circle. |A plot of the first few terms of the Taylor series of {{math|''e''<sup>''it''</sup>}} for {{math|''t'' {{=}} 2}}. ]]

Here is a proof of Euler's formula using [[Taylor series|power-series expansions]], as well as basic facts about the powers of {{mvar|i}}:<ref>{{cite book|url=https://books.google.com/books?id=PjK0F0T3NBoC&pg=PA428 |title=A Modern Introduction to Differential Equations |first=Henry J. |last=Ricardo |date=23 March 2016 |page=428|publisher=Elsevier Science |isbn=9780123859136 }}</ref>
<math display="block">\begin{align}
i^0 &= 1, & i^1 &= i, & i^2 &= -1, & i^3 &= -i, \\
i^4 &= 1, & i^5 &= i, & i^6 &= -1, & i^7 &= -i \\
&\vdots   & &\vdots   & &\vdots    & &\vdots
\end{align}</math>

Using now the power-series definition from above, we see that for real values of {{mvar|x}}
<math display="block">\begin{align}
 e^{ix} &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\[8pt]
 &= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots \\[8pt]
 &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\[8pt]
 &= \cos x + i\sin x ,
\end{align}</math>
where in the last step we recognize the two terms are the [[Taylor series#Trigonometric functions|Maclaurin series]] for {{math|cos ''x''}} and {{math|sin ''x''}}. [[Riemann series theorem|The rearrangement of terms is justified]] because each series is [[absolute convergence|absolutely convergent]].