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

When we do, the interval will be $(c-r,c+r)$. However, the boundary values for $x$, $c-r$ and $c+r$ must be tested individually for convergence. Many times, one boundary value will produce an alternating, convergent series while the other will produce a divergent, non-alternating series. As a result, intervals of convergence may not be strictly open.

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 (radius) on $|x-c|$ such that whenever $|x-c|<r$, the ratio test
$\left|{\frac {a_{n+1}}{a_{n}}}\right|<1.$

 Solution:

(a):

(b):

Final Answer:

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