When will these two series converge?

Question

Given $\lambda_n=n\lambda+\nu,\mu_n=n\mu$, (a) for which values of $\mu,\lambda,\nu$ will the series $$\sum_{n=1}^{\infty} \frac{\mu_1\cdots\mu_n}{\lambda_1\cdots\lambda_n}=\sum_{n=1}^{\infty} \frac{n! \mu^n}{\prod_{i=1}^n (i\lambda+\nu)}$$ converge and (b) for which values will the series $$\sum_{n=0}^{\infty} \frac{\lambda_0\cdots\lambda_{n-1}}{\mu_1\cdots\mu_n}=\sum_{n=0}^{\infty} \frac{\prod_{i=0}^{n-1} (i\lambda+\nu)}{n! \mu^n}$$ converge? I guess the answer should be $\mu<\lambda$ for (a) and $\mu>\lambda$ for (b) but I have no idea how to complete the proof. Thanks for help!

Answer 1

Use the ratio test. For the first series, it gives the condition

$$\lim_{n\to\infty}\Biggr| \frac{(n+1)\mu}{(n+1)\lambda+\nu}\Biggr| = \Biggr| \frac{\mu}{\lambda}\Biggr| < 1$$

with an unknown convergence if $\mu=\lambda$. The opposite condition holds with the second series. By rewriting terms, one can show that in the $\mu=\lambda$ case, the summands do not approach 0, so they diverge in both series.