Series problem: sum[n=1,infty] { (nx) * prod[k=1,n] { sin^2(k@)/(1+x^2+cos^2(k@)) } }

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slydez

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\(\displaystyle \sum_{n=1}^\infty nx\ (\ \prod_{k=1}^n \dfrac{sin^2(k\theta)}{(1+x^2+cos^2(k\theta))}\ )\)

How do I even start this problem?

Edit: This is my first time seeing the product sign, but I tried searching it up and it says that the same rules for convergence/divergence for a series applies for products. I thought of using alternating series test, since \(\displaystyle \sin^2(k\theta)\) alternates as k increases, but I would end up with \(\displaystyle \dfrac{1}{(1+x^2+cos^2(k\theta)} \). Since it is with respect to k, the lim as k approaches infinity would alternate from \(\displaystyle \dfrac{1}{1+x^2+1}\) to \(\displaystyle \dfrac{1}{1+x^2-1}\) which is divergent.

I also found out about using log but I don't know if it will work here.
 
Last edited:
slydez said:
\(\displaystyle \sum_{n=1}^\infty nx\ (\ \prod_{k=1}^n \dfrac{sin^2(k\theta)}{(1+x^2+cos^2(k\theta))}\ )\)

How do I even start this problem?
Click to expand...

Do you know what does \(\displaystyle \prod_{k=1}^n \ \) mean?
 
I'm wondering what the exact instructions for this exercise were. I note that whether the series converges or diverges depends on the values of x and \(\displaystyle \theta\). For instance, if \(\displaystyle x=\theta=1\), then the terms of the product series are bounded between 0 and 0.5. Since each term is less than one, every iteration of the product series will be smaller than the previous, eventually going to 0. Then the sum adds n times each iteration of the product series. According to my spreadsheet, the overall series does converge.

So, my best guess is that you were told to find all value(s) for x and \(\displaystyle \theta\) for which the series converges. Can you confirm if my suspicion is correct? And if it's not correct, please provide the actual instructions (if possible, quote from your book/worksheet word-for-word to help eliminate confusion).
 
slydez said:
\(\displaystyle \sum_{n=1}^\infty nx\ (\ \prod_{k=1}^n \dfrac{sin^2(k\theta)}{(1+x^2+cos^2(k\theta))}\ )\)

How do I even start this problem?

Edit: This is my first time seeing the product sign, but I tried searching it up and it says that the same rules for convergence/divergence for a series applies for products. I thought of using alternating series test, since \(\displaystyle \sin^2(k\theta)\) alternates as k increases, but I would end up with \(\displaystyle \dfrac{1}{(1+x^2+cos^2(k\theta)} \). Since it is with respect to k, the lim as k approaches infinity would alternate from \(\displaystyle \dfrac{1}{1+x^2+1}\) to \(\displaystyle \dfrac{1}{1+x^2-1}\) which is divergent.

I also found out about using log but I don't know if it will work here.
Click to expand...
What does the ratio test look like?
 
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