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This was asked at an oral examination.

Does the series $\displaystyle \sum _{k\geq1}\frac{\sin\left(\sqrt{k}\right)}{k}$ converge ?

After playing with Mathematica, it's very likely it converges, but slowly (sort of oscillating).

To actually prove convergence, summation by part is useless since $\displaystyle \sum _{k\geq1}\sin\left(\sqrt{k}\right)$ diverges.

Any suggestion is appreciated.

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  • $\begingroup$ It does converge by the integral test. $\endgroup$ – Alex Schiff Jun 9 '14 at 19:46
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    $\begingroup$ @AlexSchiff, the terms are not nonnegative nonincreasing. $\endgroup$ – Christopher A. Wong Jun 9 '14 at 19:59
  • $\begingroup$ Whoopsie daisy, not enough coffee this morning. :-P $\endgroup$ – Alex Schiff Jun 9 '14 at 20:06
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The convergence of $\int_1^\infty\frac{\sin\sqrt t}tdt$ can be deduced by a substitution $s=\sqrt t$ and the convergence of $\int_1^\infty\frac{\sin s}sds$. Define $$a_k:=\int_k^{k+1}\frac{\sin\sqrt t}{t}\mathrm dt-\frac{\sin\sqrt k}k$$ and $g(x):=\frac{\sin(\sqrt x)}x$. Since $$g'(x)=\frac{\cos(\sqrt x)}{x^{3/2}}-\frac{\sin(\sqrt x)}{x^2}$$ and by the mean value theorem, $a_k=g'(x_k)$ for some $x_k\in [k,k+1)$, we obtain $$|a_k|\leqslant \frac 2{k^{3/2}},$$ hence $\sum_{k\geqslant 1}|a_k|$ is convergent.

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  • $\begingroup$ There's one thing I don't get: how did you apply the mean value theorem ? $\endgroup$ – Gabriel Romon Jun 9 '14 at 20:13
  • $\begingroup$ Explain why $a_k = g'(x_k)$ by MVT? $a_k$ is not equal to $g(k+1) - g(k)$. $\endgroup$ – Christopher A. Wong Jun 9 '14 at 20:13
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    $\begingroup$ @ChristopherA.Wong The integral $\int_k^{k+1}\frac{\sin\sqrt t}{t}\mathrm dt$ is $g(y_k)$ for some $y_k\in [k,k+1]$, then $|g(y_k)-g(k)|\leqslant \sup_{t\in [k,k+1]}|g'(t)|$. $\endgroup$ – Davide Giraudo Jun 9 '14 at 20:17
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$\sum_{k=1}^{n}\dfrac{\sin(\sqrt{k})}{k}$ plotted against $C-\dfrac{2\cos(\sqrt{k-\sqrt{\pi}})}{\sqrt{k+\sqrt{\pi}}}$, where $C$ is calculated in Mathematica numerically with fact that $\sum_{k=1}^{\infty}\dfrac{\sin(\sqrt{k})}{k}=\sum_{k=1}^{\infty}\dfrac{e^{i\sqrt{k}}}{k}$:

enter image description here

c = Quiet[N[Im[Sum[E^(I Sqrt[k])/k, {k, 1, Infinity}]]]];
Show[Plot[Sum[Sin[Sqrt[k]]/k, {k, 1, n}], {n, 0, 100}, 
GridLines -> {{}, {c}}, GridLinesStyle -> Darker[Green], 
PlotStyle -> {Darker[Green], Thick}], 
Plot[c - 2 Cos[Sqrt[k - Sqrt[Pi]]]/Sqrt[k + Sqrt[Pi]], {k, 0, 100}, 
PlotStyle -> {Darker[Blue], Thin}, PlotRange -> All]]

and for larger $k$:

enter image description here

Quiet[N[Im[Sum[E^(I Sqrt[k])/k, {k, 1, Infinity}]]]]

then gives $C\approx1.7156717726570607\dots$.

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  • $\begingroup$ Thanks for portraying the series. Does WolframAlpha/Mathematica suggests a closed form for the limit ? (Can you copy-paste it with at least 5 digits precision ?) $\endgroup$ – Gabriel Romon Jun 10 '14 at 13:49
  • $\begingroup$ @ G.T.R. - Please see update. $\endgroup$ – martin Jun 11 '14 at 6:10
  • $\begingroup$ Interesting. WolframAlpha finds multiple possible closed Forms for $C$ m.wolframalpha.com/input/?i=1.7156717726570607&x=8&y=0 $\endgroup$ – Gabriel Romon Jun 11 '14 at 8:01
  • $\begingroup$ Hmmm ... just typed in completely random number (1.254635278541656) & also came up with possible closed forms ;) $\endgroup$ – martin Jun 11 '14 at 8:57
  • $\begingroup$ Can you get 20 digits precision ? There will be less possible closed Forms then. $\endgroup$ – Gabriel Romon Jun 11 '14 at 9:25

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