Checking series for convergence

Lambda96
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Homework Statement
Check whether the series ##\sum\limits_{k=1}^{\infty} \frac{\Bigl( 1 + \frac{1}{k} \Bigr)^{k^2}}{3^{k}}## converges
Relevant Equations
All convergence criteria are allowed
Hi,

I am having problems with task d)

Bildschirmfoto 2023-12-06 um 18.30.36.png


I now wanted to check the convergence using the quotient test, so ## \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1##

I have now proceeded as follows:

##\frac{a_{n+1}}{a_n}=\frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{3^{k+1}} \cdot \frac{3^{k}}{\Bigl( 1 + \frac{1}{k} \Bigr)^{k^2}}##

##\frac{a_{n+1}}{a_n}=\frac{1}{3} \frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{\Bigl( 1 + \frac{1}{k} \Bigr)^{k^2}}##

##\frac{a_{n+1}}{a_n}=\frac{1}{3} \Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2} \cdot \Bigl( 1 + \frac{1}{k} \Bigr)^{-k^2}##

Unfortunately I can't get any further now, if I form the limit ##\lim_{k\to\infty} \frac{1}{3} \Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2} \cdot \Bigl( 1 + \frac{1}{k} \Bigr)^{-k^2} ## with Mathematica, ##\frac{e}{3}## must come out, unfortunately I don't know how I can show this with my expression, or should I have used a different criterion for the task?
 
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Look at <br /> \lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right| = \exp\left( \lim_{k \to \infty} (\ln |a_{k+1}| - \ln |a_{k}|) \right) and use <br /> \ln(1 + x) = x - \frac12 x^2 + O(x^3), \quad |x| &lt; 1
 
Last edited:
Lambda96 said:
Homework Statement: Check whether the series ##\sum\limits_{k=1}^{\infty} \frac{\Bigl( 1 + \frac{1}{k} \Bigr)^{k^2}}{3^{k}}## converges
Relevant Equations: All convergence criteria are allowed

Hi,

I am having problems with task d)

View attachment 336770

I now wanted to check the convergence using the quotient test, so ## \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1##

I have now proceeded as follows:

##\frac{a_{n+1}}{a_n}=\frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{3^{k+1}} \cdot \frac{3^{k}}{\Bigl( 1 + \frac{1}{k} \Bigr)^{k^2}}##

##\frac{a_{n+1}}{a_n}=\frac{1}{3} \frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{\Bigl( 1 + \frac{1}{k} \Bigr)^{k^2}}##

##\frac{a_{n+1}}{a_n}=\frac{1}{3} \Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2} \cdot \Bigl( 1 + \frac{1}{k} \Bigr)^{-k^2}##

Unfortunately I can't get any further now, if I form the limit ##\lim_{k\to\infty} \frac{1}{3} \Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2} \cdot \Bigl( 1 + \frac{1}{k} \Bigr)^{-k^2} ## with Mathematica, ##\frac{e}{3}## must come out, unfortunately I don't know how I can show this with my expression, or should I have used a different criterion for the task?
Seems you could use that
##Lim_{k\rightarrow \infty}(1+\frac{1}{k})^k =e##
 
WWGD said:
Seems you could use that
##Lim_{k\rightarrow \infty}(1+\frac{1}{k})^k =e##

How does that generalize to the case where the exponent is k^2 rather than k?
 
Thank you pasmith and WWGD for your help 👍👍, I have now used the root test, which allowed me to reduce the expression to ##\frac{(1+\frac{1}{k})^k}{3}## and which corresponds to the limit ##\lim_{k \to \infty}## to ## \frac{e}{3}##
 
pasmith said:
How does that generalize to the case where the exponent is k^2 rather than k?
Well, Lambda96 used it to solve his problem. I only provide hints, as per PF policy.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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