# Functions : Radius of Convergence and Approximations

Please see the attached file for the fully formatted problems.

Let f be the function defined by f(x) = sigma (starting at n=1 ending at infinity)

xnnn /(3n n!) for all values of x for which the series converges.

a) Find the radius of convergence of this series

b) Use the first three terms of this series to approximate f(-1)

c) Estimate the amount of error involved in the approximation in part b. Justify your answer

https://brainmass.com/math/graphs-and-functions/functions-radius-convergence-approximations-17185

#### Solution Preview

a)

an=x^n*n^n /(3^n*n!)

an+1=x^(n+1)*(n+1)^(n+1) /(3^(n+1)*(n+1)!)

|an+1/an|=|1/3*x*(n+1)^n/(n^n)|

lim|1/3*x*(n+1)^n/(n^n)|=|ex/3|

n--->infinity

Note ...

#### Solution Summary

The radius of convergence and an approximation of a function are found.

$2.19