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 + (9x^2)/(2!) + (27x^3)/(3!) + \cdots = \sum_(n=0)^\infty (3^nx^n)/(n!) = \sum_(n=0)^\infty ((3x)^n)/(n!)`

Recall that

`\displaystyle e^x = \sum_(n=0)^\infty (x^n)/(n!)`

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