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