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I'm trying to find, whether the series $$\sum_{n = 1}^{+\infty}nx\prod_{k = 1}^{n}\frac{\sin^2(k\alpha)}{1 + x^2 + \cos^2(k\alpha)}.$$ I'm trying to find it using the ratio test. In order to determine convergence of series I need to evaluate limit $$\lim_{n \to +\infty} \frac{n+1}{n}\cdot\frac{\sin^2(\alpha +\alpha n)}{1 + x^2 + \cos^2(\alpha + \alpha n)}$$ and I've tried to use identities, expansions and so, but I can't find the way. Where to proceed?

Thanks!

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  • $\begingroup$ Could you post your original question? $\endgroup$
    – DHMO
    Commented Apr 14, 2017 at 17:42
  • $\begingroup$ That limit does not converge. $\endgroup$
    – DHMO
    Commented Apr 14, 2017 at 17:43
  • $\begingroup$ This seems to have oscillatory value (i.e. limit doesn't exist), can you post original problem? $\endgroup$ Commented Apr 14, 2017 at 17:43
  • $\begingroup$ "whether the series ..." that needs completing $\endgroup$
    – zhw.
    Commented Apr 14, 2017 at 18:30

1 Answer 1

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Your goal in applying the ratio test is to prove that $|a_{n+1}/a_n|\le k<1$ for all sufficiently large $n$ and some fixed $k$. (Baby Rudin 3.34.) You don't need for $|a_{n+1}/a_n|$ to converge. Oscillation is fine so long as it's oscillation in an interval $[0,k]$ with $k<1$.

In this case, for large $n$ we have $(n+1)/n$ arbitrarily close to $1$, while $$\frac{\sin^2(\alpha +\alpha n)}{1 + x^2 + \cos^2(\alpha+\alpha n)}\le {1\over 1+x^2}$$ The ratio test is therefore met for $n$ so large that ${n+1\over n}\cdot {1\over 1+x^2}<1$. This will happen for sufficiently large $n$ whenever $x\ne 0$.

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  • $\begingroup$ You might consider stating that all we need is $\limsup_{n+1}\left|\frac{a_{n+1}}{a_n}\right|<1$ to assure convergence. You've tacitly shown this, but you might just consider explicitly stating it. (+1) for the answer. -Mark $\endgroup$
    – Mark Viola
    Commented Apr 14, 2017 at 19:50

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