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# Functions : Radius of Convergence and Approximations

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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

##### Solution Summary

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

##### 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

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