Question

Find the radius and the interval of convergence for:

Answer 1

\[\sum_{k=1}^{\infty} [ \frac { (-1)^k (\frac{ x+1 }{ 2 }) ^k}{ k } ]\]

Answer 2

gotta wrap it in delimiters to code out right

thats better

Answer 3

how far have you gotten?

Answer 4

might want to go ahead an jsut replace (x+1) by g(x), or just g so that the end doesnt confuse us

Answer 5

Apply the ratio test:
\[\lim_{k\to\infty}\left|\frac{(-1)^{k+1}\left(\frac{x+1}{2}\right)^{k+1}}{k+1}\cdot\frac{k}{(-1)^{k}\left(\frac{x+1}{2}\right)^k}\right|\\
\lim_{k\to\infty}\left|\frac{x+1}{2}\right|\frac{k}{k+1}\\
\left|\frac{x+1}{2}\right|\]
By the ratio test's conditions, the series will converge if ... ?

Answer 6

i believe im doing simple algebra mistakes on this because i ended with \[\lim_{k \rightarrow \infty} \left|\frac{ -1 \frac{ x+1 }{ 2} }{ k+1 } \right|\]

Answer 7

also in the numerator there is a k

Answer 8

Well it looks like you have the same thing as me. Since \(k\to\infty\), you're dealing with positive \(k\), so you can remove the absolute value bars. Also, \(|-1|=1\), so that's why the \((-1)^\cdots\) disappear.

And since the limit takes \(k\) into consideration, you can pull out the term containing \(x\).