009C Sample Final 3, Problem 6

Consider the power series

$\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{n+1}}{n+1}}$

a) Find the radius of convergence of the above power series.

b) Find the interval of convergence of the above power series.

c) Find the closed formula for the function $f(x)$ to which the power series converges.

d) Does the series

$\sum _{n=0}^{\infty }{\frac {1}{(n+1)3^{n+1}}}$

converge? If so, find its sum.

Foundations:

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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