an infinite series that i couldn't figured out how to sum up

Question

I have the series which is $$\sum_{n=0}^{\infty} 2^{-n(x-1)}$$ and from the ratio test it converges for all $x\geq 2$ but how can i find the general sum of the series wrt $x$

Answer 1

Note, that you've in general:

$$\sum_{\text{n}\ge0}x^\text{n}\tag1$$

Which is the geometric series. In your case we have:

$$x=2^{1-\text{k}}\tag2$$

Answer 2

$$f(x) = \sum_{n=0}^{\infty} 2^{-n(x-1)} = \sum_{n=0}^{\infty} \left(2^{x-1}\right)^{-n} = \frac{1}{1- 2^{1-x}}$$

Answer 3

Hint

$$\sum_{n=0}^{+\infty} \frac{1}{2^{n(x-1)}}=\sum_{n=0}^{+\infty} \frac{1}{(2^{x-1})^n}=\sum_{n=0}^{+\infty} \frac{1}{(2^{x-1})^n} =\sum_{n=0}^{+\infty} \left(\frac{1}{2^{x-1}} \right)^n= \sum_{n=0}^{+\infty} y^n$$

where $y = \frac{1}{2^{x-1}}$.

Can you continue from here?