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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (c-r,c+r)} . However, the boundary values for $x$, $c-r$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x-c|} such that whenever Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x-c|<r} , the ratio test
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|<1.}

 Solution:

(a):

(b):

Final Answer:

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