Can this sum of series have a closed form representation?

Question

Can this summation be represented as a closed form or reduced to some other easily calculable series.
$\sum_{i=0}^n {x^{p^i}}$

Answer 1

--------------I edited the comment to extend my answer--------------------

I assume your goal is to understand the convergence/divergence of $$\sum_{n=1}^{\infty} x^{p^n}$$.If so, then this approach works for any integer $|p|\geq1$ and for any $x>0$ real number.

Suppose,you have $$\sum_{n=1}^{\infty} x^{p^n}$$, for some $|p|\geq1$ & $x>0$.

Let $y= x^{p^n}$. Therefore, $log(y)=p^n logx$, for all $n$.We can consider the series of the logarithms.i.e: $$\sum_{n=1}^{\infty} log(y)=\sum_{n=1}^{\infty} p^n logx=log x(\sum_{n=1}^{\infty} p^n)$$.

Now,we can can come to conclusion about $$\sum_{n=1}^{\infty} x^{p^n}$$. For any $|p|\geq 1$ & $x>0$ , the series $$\sum_{n=1}^{\infty} x^{p^n}$$ diverges since $$\sum_{n=1}^{\infty} log(y)$$ diverges.

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Alternatively, you could use the Euler-Maclaurin integration formula to estimate the finite sum. But, it is not a pretty one(For a function $f$ that is infinitely differentiable)

This is the full Euler-Maclaurin formula with no remainder term & $B_k$ is the $k^{th}$ Bernoulli number.