Series proof for $e^x$.

Question

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.

Answer 1

Use any definition of the exponential function you want, then prove that it is equivalent to that infinite sum. Example: $$e^x=\lim_{n\to\infty}\left(1+\dfrac xn\right)^n,$$ we just have to prove the following: $$\lim_{n\to\infty}\left(1+\dfrac xn\right)^n=\sum_{n=0}^\infty \frac{x^n}{n!}.$$ You can find out the proof of the equivalence of definitions of $e^x$ in this article.