radius and interval of convergence for series

[djdavis2k]

Find the radius of convergence and the interval of convergence for each of the series listed below:

a.) sigma notation from (k=1 to infinity) of ((5^k)/k^2))*(x^k)

b.) sigma notation from (n=1 to infinity) of {((-1)^n)*(x^2*n)}/(2*n)!

c.) sigma notation from (k=1 to infinity) of {(ln k)(x-1)^k}/(k)

i know radius of convergence is the -1/L to 1/L value for the ratio test, and the roots test can be used as well... but I need help applying it to these sigma notations ... assistance here would be greatly appreciated

 

[galactus]

Find the radius of convergence and the interval of convergence for each of the series listed below:

a.)\(\displaystyle \sum_{k=1}^{\infty}\frac{5^{k}}{k^{2}}\cdot x^{k}\)

Using the ratio test:

\(\displaystyle \lim_{k\to \infty}\left|\frac{5^{k+1}x^{k+1}}{(k+1)^{2}}\cdot\frac{k^{2}}{5^{k}x^{k}}\right|\)

\(\displaystyle =\lim_{k\to \infty}\left|\frac{5k^{2}}{(k+1)^{2}}x\right|=5|x|\)

The series converges if \(\displaystyle -1<5x<1\)

\(\displaystyle \frac{-1}{5}<x<\frac{1}{5}\)

Converges if \(\displaystyle |x|<\frac{1}{5}\) and diverges if \(\displaystyle |x|>\frac{1}{5}\)

If \(\displaystyle x=\frac{-1}{5}, \;\ \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k^{2}}\) converges;

If \(\displaystyle x=\frac{1}{5}, \;\ \sum_{k=1}^{\infty}\frac{1}{k^{2}}\) converges.

Radius of convergence is 1/5, interval of convergence is \(\displaystyle [\frac{-1}{5}, \frac{1}{5}]\)

There is a nice spelled out example. Can you get the others OK?. Let me know of you remain stuck, but let's see some of your attempt.

 

[djdavis2k]

for b.) if i use the ratio test

i get the limit of n approaches infinity of absolute value of (x^(2n+1)/(2n+1)!)*((2n)!/(x^2n))

which is lim of n approach infinity of (x)/(2n+1)= 0

how do i find radius of convergence and integral of converge there...


for c.)

if i apply the ratio test i get limit of k approaches of infinity of (ln(k+1)*(x-1)*k)/((ln k)*(k+1))

how do i simiplify this to get an overall limit.. do i apply l'hopital's rule and if so how do i go about doing so.. and how to i use this info to find the radius of converge and interval of convergance

sorry for disturbing you... thanks for your help so far

 

[galactus]

For b, we have \(\displaystyle {\rho}=\lim_{k\to \infty}\frac{|x|^{2}}{(2k+2)(2k+1)}=0\)

radius of convergence is \(\displaystyle +\infty\) and interval of convergence is \(\displaystyle (-\infty, \;\ +\infty)\)

Gotta go. Be back later.

 

[johnq2k7]

Find the radius of convergence and the interval of convergence for each of the series listed below:

a.)\(\displaystyle \sum_{k=1}^{\infty}\frac{5^{k}}{k^{2}}\cdot x^{k}\)

Using the ratio test:

\(\displaystyle \lim_{k\to \infty}\left|\frac{5^{k+1}x^{k+1}}{(k+1)^{2}}\cdot\frac{k^{2}}{5^{k}x^{k}}\right|\)

\(\displaystyle =\lim_{k\to \infty}\left|\frac{5k^{2}}{(k+1)^{2}}x\right|=5|x|\)

The series converges if \(\displaystyle -1<5x<1\)

\(\displaystyle \frac{-1}{5}<x<\frac{1}{5}\)

Converges if \(\displaystyle |x|<\frac{1}{5}\) and diverges if \(\displaystyle |x|>\frac{1}{5}\)

If \(\displaystyle x=\frac{-1}{5}, \;\ \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k^{2}}\) converges;

If \(\displaystyle x=\frac{1}{5}, \;\ \sum_{k=1}^{\infty}\frac{1}{k^{2}}\) converges.

Radius of convergence is 1/5, interval of convergence is \(\displaystyle [\frac{-1}{5}, \frac{1}{5}]\)

There is a nice spelled out example. Can you get the others OK?. Let me know of you remain stuck, but let's see some of your attempt.

or b.) if i use the ratio test

i get the limit of n approaches infinity of absolute value of (x^(2n+1)/(2n+1)!)*((2n)!/(x^2n))

which is lim of n approach infinity of (x)/(2n+1)= 0

so is it -1 equals x/2n+1 and 1

so -2n-2 equals x or 2n+2

how do i find radius of convergence and integral of converge there...


for c.)

if i apply the ratio test i get limit of k approaches of infinity of (ln(k+1)*(x-1)*k)/((ln k)*(k+1))

is it -1 equals (ln(k+1)*(x-1)*k)/((ln k)*(k+1)) or 1

how do i simiplify this to get an overall limit.. do i apply l'hopital's rule and if so how do i go about doing so.. and how to i use this info to find the radius of converge and interval of convergance

sorry for disturbing you... thanks for your help so far