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:
Ratio Test
Let  $\sum a_{n}$  be a series and  $L=\lim _{n\rightarrow \infty }{\bigg |}{\frac {a_{n+1}}{a_{n}}}{\bigg |}.$
Then,
If  $L<1,$  the series is absolutely convergent.
If  $L>1,$  the series is divergent.
If  $L=1,$  the test is inconclusive.

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

(d)

Step 1:

Step 2:

Final Answer:
(a)
(b)
(c)
(d)

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