Question

Find the function to which the given series converges within its interval of convergence. Use exact values. 1 3 x 9 x 2 2 ! 27 x 3 3 ! 81 x 4 4 ! 243 x 5 5 !

Answer 1

It looks like the series could be

$\displaystyle 1 + 3x + \frac{9x^2}{2!} + \frac{27x^3}{3!} + \cdots = \sum_{n=0}^\infty \frac{3^nx^n}{n!} = \sum_{n=0}^\infty \frac{(3x)^n}{n!}$

Recall that

$\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$

It follows that the given series is the power series expansion for $\boxed{e^{3x}}$.