Is f(x) Representable as a Special Function?

[pierce15]

Hi, does anyone know if this function:

$$f(x) = \sum_{k=1}^\infty \frac{(-1)^n}{x^{2k}}$$

is representable as an elementary or already defined special function? Thanks

 

[pierce15]

This function can be broken up as:

$$f(x) = \sum_{k=1}^\infty x^{4k} - \sum_{n=1}^\infty x^{4n-2}$$

Any ideas?

 

[pierce15]

I think I've got it: the above expression is equal to

$$\frac{1}{x^4 - 1} - \frac{ x^2} {x^4 - 1} = - \frac{1}{1 + x^2}$$

does that look ok?

 

[mathman]

I think I've got it: the above expression is equal to

$$\frac{1}{x^4 - 1} - \frac{ x^2} {x^4 - 1} = - \frac{1}{1 + x^2}$$

does that look ok?

Answer is correct. It is a geometric series, ratio = -1/x².

 

[pierce15]

Oh, right. I'm an idiot for not seeing that to begin with