help with infinite sum

[sand veldt]

I was doing integration of the floor function and i got to the sum, i dont know how to evaluate it, f(1) and f(0) are just constants.

$$\sum_{k=1}^{\infty} \frac{\sin(2\pi k f(1)) - 2\pi k \cos(2\pi k f(1)) - \sin(2\pi k f(0))}{4\pi^3 k^3}$$

[spice ingot]

alaska v.2

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[quasi kestrel]

Looks like a Fourier series.

[polar vapor]

...what's with all the sine and cosine stuff? What interval are you integrating over?

[sand veldt]

I was integrating $\int_0^1 \left\lfloor \frac{1}{f(x)} \right\rfloor x \dd x$

ok there

[spice ingot]

alaska v.2

[quasi kestrel]

Oh.

I did derive a formula for integrals involving floor functions.

[sand veldt]

oh dam really

[quasi kestrel]

Though, I believe it was for f(⌊x⌋), not ⌊f(x)⌋.

[sand veldt]

oh

[quasi kestrel]

Integrals of the latter are usually hard to evaluate.

[opal mulch]

Hm maybe trying to get the fourrier series expansion of $f: x \to x^{3}-\pi^{2}x$ on the interval $]-\pi; \pi[$ and repeated on a $2 \pi$ period

[spice ingot]

Rotor 😑

[opal mulch]

oh nvermind you get $f(x)=12\sum_{n=1}^{+\infty} \frac{ (-1)^{n}}{n^{3}} sin(nx)$

[spice ingot]

Rotor 😑

[quasi kestrel]

I kinda forgot what functions have simple Fourier coefficients. I think that's worth investigating.

I'll do that in the morning.

[opal mulch]

I’ll do the same I think I haven’t done fourrier analysis in a long time so this could de rust me

[opal mulch]

hmm im testing some different fourrier series expansions and this is unrelated, but and i may have found a proof of what $\sum_{n=1}^{+\infty} \frac{1}{n^{2}+a^{2}}$ is for a>0 using fourrier analysis

[spice ingot]

Rotor 😑

[quasi kestrel]

Here's what I got.

This doesn't cover all the simple cases, but it should be enough to evaluate that sum.

[opal mulch]

I see, cool!

fourrier series are pretty interesting