009C Sample Midterm 3, Problem 5

Find the radius of convergence and the interval of convergence of the series.

(a) (6 points)      ${\displaystyle \sum _{n=0}^{\infty }}{\frac {(-1)^{n}x^{n}}{n+1}}.$

(b) (6 points)      ${\displaystyle \sum _{n=0}^{\infty }}{\frac {(x+1)^{n}}{n^{2}}}.$

Foundations:
When we are asked to find the radius of convergence, we are given a series where
$a_{n}=f(x-c)\cdot g(n)$
where $f$ and $g$ are functions of $x$ and $n$ respectively, and $c$ is a constant (frequently zero). We need to find a bound on $|x-c|$ such that whenever $|x-c|<L$, the ratio test
$\left|{\frac {a_{n+1}}{a_{n}}}\right|$
is satisfied. When we do, the interval will be $(c-L,c+L)$. However, the boundary values for $x$, $c-L$ and $c+L$ must be tested individually for convergence. Most often, one will produce an alternating, convergent series while the other will produce a divergent, non-alternating series.

 Solution:

(a):

(b):

Final Answer:
The series for (a) is convergent, while the series (b) is divergent.

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