Jtdylktuy

As the title says, I need to calculate the sum of the function series $$sum_{n=1}^{infty}frac{cos^3(nx)}{n^2}$$ so that I can find out if the series does simple or uniform converge to something.

Usually, for problems like this I need to write the sum like a difference of sums so when we write the sums, some terms will go away, but I can't think of a way to write this one.

Can you help me?

UPDATE

I managed to show that the series is uniform convergent even if I do not know to who it converges. Still, I wonder if there is any way to calculate the sum of the series.

$$sum_{ngeq 1}frac{cos(nx)}{n^2} = text{Re},text{Li}_2(e^{ix})$$
is a periodic and piecewise-parabolic function, as the primitive of the sawtooth wave $sum_{ngeq 1}frac{sin(nx)}{n}$.

Since $cos^3(nx)=frac{1}{4}cos(3nx)+frac{3}{4}cos(nx)$ your function is a piecewise-parabolic function too.

In explicit terms it is a $2pi$-periodic, even function which equals $frac{3}{4}left(x-frac{pi}{3}right)left(x-frac{2pi}{3}right)$ over $[0,2pi/3]$ and $frac{3}{4}left(x-frac{2pi}{3}right)left(x-frac{4pi}{3}right)$ over $[2pi/3,pi]$.

This is very simple to derive by interpolation once the original series is evaluated at $xinleft{0,frac{pi}{3},frac{2pi}{3},pi,frac{4pi}{3},frac{5pi}{3}right}$.

$newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
newcommand{dd}{mathrm{d}}
newcommand{ds}[1]{displaystyle{#1}}
newcommand{expo}[1]{,mathrm{e}^{#1},}
newcommand{ic}{mathrm{i}}
newcommand{mc}[1]{mathcal{#1}}
newcommand{mrm}[1]{mathrm{#1}}
newcommand{pars}[1]{left(,{#1},right)}
newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
newcommand{root}[2]{,sqrt[#1]{,{#2},},}
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"As the title says, I need to calculate the sum of the function series..."

begin{equation}
bbx{mbox{It's clear that}
sum_{n=1}^{infty}{cos^{3}pars{nx} over n^{2}} = {pi^{2} over 6phantom{^{2}}} mbox{when}
braces{verts{x} over 2pi} = 0}
end{equation}

With
Jonqui$grave{mathrm{e}}$re Inversion Formula, it's found
begin{align}
&left.bbox[10px,#ffd]{sum_{n=1}^{infty}{cos^{3}pars{nx} over n^{2}}},rightvert
_{ verts{x}/pars{2pi} not= 0}
\[5mm] = &
bbx{{3 over 4},bracks{pi^{2},mrm{B}_{2}pars{braces{verts{x} over 2pi}}} +
{1 over 4},bracks{pi^{2},
mrm{B}_{2}pars{braces{3braces{verts{x} over 2pi}}}}}
end{align}

$ds{mrm{B}_{n}}$ is a
Bernoulli Polynomial.
Note that
$ds{mrm{B}_{2}pars{z} = z^{2} - z + {1 over 6}}$.