Find sum of this convergent series

Question

find the sum of the infinite series

$\sum_{i=0}^\infty\frac{2^i}{n^{(2^i)}}$ for $n>1$

I tried the following

$\frac{1}{n}+\frac{2}{n^2}+\frac{4}{n^4}+...=k$

$\frac{1}{n}+\frac{2}{n^2}(1+\frac{2}{n^2}+...)=k$

But it is not helping because the denomenator of $8$ will become $n^6$

Answer 1

There is no expression for this series in "standard" functions. The series $f(z) = \sum_{i=0}^\infty z^{2^i}$ is a well-known example of a lacunary series (but with no generally accepted name), and this is $ f'(1/n)/n$.