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I am told to analyze the convergence of $sum^infty _{n=1}{frac{1}{p^n - q^n}}, 0lt qlt p$ depending on the values of $p$ and $q$. The only thing I have concluded is that ovbiously, the series is a positive term series. I do not know which criteria to use in this case. Thanks in advance.

Observe that
$$sum_{n=1}^inftyfrac{1}{p^n-q^n}=frac{1}{p-q}sum_{n=1}^inftyfrac{1}{p^{n-1}+p^{n-2}q+cdots+pq^{n-2}+q^{n-1}}$$
We can apply the comparison test to $dfrac{1}{p^{n-1}}$ and $dfrac{1}{q^{n-1}}$ to see that the series converges whenever $p>1$ or $q>1$.

If both $pleq 1$ and $qleq 1$, then it is not the case that $dfrac 1{(p^n-q^n)}to 0$, hence the series diverges.

Write the general term $a_n$ of the series as
$$
0lefrac{m}{p^n}leq a_n=frac{1}{p^n(1-(q/p)^n)}leq frac{M}{p^n}
$$
since $frac{1}{1-(q/p)^n}to 1$ for some $m, M>0$. The above inequality should allow you to deduce convergence depending on whether $p>1$ or $p<1$.

For the case with $p=1$, the general term of the series is
$$
a_n=frac{1}{1-q^n}
$$
What can we say about $lim_{nto infty} a_n$ if $q<1$.