I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum:

$$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$

or, an alternate form:

$$\sum_{n=1}^{\infty} \frac{n-1}{n^{a+1}-1}$$

For low values of $a$, Wolfram Alpha gives a closed form in terms of the polygamma function function of order $0$, so I am wondering if there is a general closed form in terms of $a$.

I asked this question on MSE, but have not received answers yet. I realize I did ask it recently there, but I also wanted to ask it here just to see if anyone had seen this sum before.

I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum:

$$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$

or, an alternate form:

$$\sum_{n=1}^{\infty} \frac{n-1}{n^{a+1}-1}$$

For low values of $a$, Wolfram Alpha gives a closed form in terms of the polygamma function function of order $0$, so I am wondering if there is a general closed form in terms of $a$.

I asked this question on MSE, but have not received answers yet. I realize I did ask it recently there, but I also wanted to ask it here just to see if anyone had seen this sum before.