Exercise 2, § 4

Let $(p_n)$ be an increasing sequence of numbers $> 0$ tending to $+\infty$.

a) If the ratio $p_n / p_{n-1}$ tends to 1 as $n$ increases indefinitely show that

(apply prop. 2 of V, p. 237).

b) Without the hypothesis on the ratio $p_n / p_{n-1}$, show that the series with general term $(p_n - p_{n-1}) / p_n$ always has an infinite sum (distinguish the two cases according to whether $p_n / p_{n-1}$ tends to 1 or not).

c) Show that for every $\rho > 0$ the series with general term $(p_n - p_{n-1}) / p_n p_{n-1}^\rho$ converges (compare to the series with general term $\frac{1}{p_{n-1}^\rho} - \frac{1}{p_n^\rho}$).