How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ?
I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number }
How to proceed ?
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Use Dirichlet's test. If $\{a_n\}$ is a sequence whose partial sums are bounded, and $\{b_n\}$ is a decreasing sequence with $\lim b_n=0$, then $\sum a_nb_n$ converges.
Tim Raczkowski
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So is taking $a_n = cos(n) $ and $b_n = \frac {1}{\sqrt n}$ would work ? – Apr 06 '15 at 15:36
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Yes, see mich95's post below. – Tim Raczkowski Apr 06 '15 at 15:41
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How can I prove ∑cos(n) is bounded. Can It be done using cos as real part of complex number as I showed in question ? – Apr 06 '15 at 18:24
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I think you can. One easier way is to use some trig formulas. $2\sin a \cos b= \sin(x+y)-\sin(x-y)$ Let $a=\frac{1}{2}$ and sum over b as b ranges from 1 to n. – mich95 Apr 06 '15 at 19:02
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Notice that $\sum\limits_{k=1}^{n} \cos k = \frac{\sin(n+\frac{1}{2})-\sin\frac{1}{2}}{2\sin\frac{1}{2}}$. Hence the sum is bounded. And since $\frac{1}{\sqrt{n}}$ is decreasing to zero, then the series converge.
Martin Sleziak
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mich95
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How can I prove $\sum cos(n)$ is bounded. Can It be done using cos as real part of complex number as I showed in question ? – Apr 06 '15 at 18:03