The Sum of a series with exponents

islandboy401
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Homework Statement



Problem: Indicate whether the series converges or diverges. If it converges, find its sum.

THE SERIES:

sum.JPG




Homework Equations



The ratio test and w/e equation is used to find the sum of this particular series

The Attempt at a Solution



I was able to find that the series converges, using the ratio test. However, I cannot find the sum of the series. I do not see any way in which I could manipulate the geometric series, or anything like that. Could someone please enlighten me on how to find the sum of this series, or any series in such a form.
 
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I don't think you need to do a lot of manipulation on a geometric series to find the sum. It should be pretty straightforward. Just factor out the k=1 term.
 
I have factored out the k=1 term...yet, I still do not know exactly what to do. I wrote out the series for both the numerator and the denominator, yet I do not see what to do next.

Here is my work:

sumone.jpg


Another hint at the problem solving process will be greatly appreciated.

Thanks.
 
you could just write 2k+1=2k*2 and 5k-1=5k*5-1
 
Thanks for all the hints...but now, I have another problem. I obtained an answer of 50/3...however, the solution booklet says the answer is 20/3...

Here is my work:

sumagain.jpg


Is this a typo in the manual, or is my answer truly wrong? If so, please tell me where I messed up.
 
The problem is that your sum is going from 1 to infinity, not zero to infinity. The formula for the geometric series that you used requires that the sum goes from zero to infinity:

\sum_{k=0}^{\infty}ax^k=\frac{a}{1-x}

But

\sum_{k=1}^{\infty}ax^k=\frac{a}{1-x}-a
 
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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