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
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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