Question

Confirm the integration formula by integrating the appropriate Maclaurin series term by term.

\int e^xdx=e^x +C

Answer 1

\[e^x=\sum_{n=0}^\infty \frac{x^n}{n!}~~\implies~~\int e^x\,dx=\sum_{n=0}^\infty \frac{x^{n+1}}{(n+1)n!}+C=\sum_{n=0}^\infty \frac{x^{n+1}}{(n+1)!}+C\]
This almost matches the sum on the left except for \(n+1\) in place of \(n\). How can you adjust this?