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Question Number 134061 by benjo_mathlover last updated on 27/Feb/21

If Σ a_{n} is a convergent series of

nonnegative terms, what can be

said about Σ a_{n} . a_{n + 1} ?

(a) always converges

(b) always diverges

(c) may converges or diverge

Answered by mathmax by abdo last updated on 27/Feb/21

Σ a_{n} a_{n + 1} ⩽ {(Σ a_{n}^{2})}^{\frac{1}{2}} {(Σ a_{n + 1}^{2})}^{\frac{1}{2}} (cauchy shwarz) the series (Σ a_{n}^{2}) and Σ a_{n + 1}^{2} cv \Rightarrow

Σ a_{n} a_{n + 1} converges