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](https://www.tinkutara.com/question/Q134061.png)
$$\mathrm{If}\:\Sigma\:\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{series}\:\mathrm{of} \\ $$$$\mathrm{nonnegative}\:\mathrm{terms},\mathrm{what}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{said}\:\mathrm{about}\:\Sigma\:\mathrm{a}_{\mathrm{n}} .\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{always}\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{always}\:\mathrm{diverges} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{may}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverge} \\ $$
Answered by mathmax by abdo last updated on 27/Feb/21
![Σa_n a_(n+1) ≤(Σa_n ^2 )^(1/2) (Σa_(n+1) ^2 )^(1/2) (cauchy shwarz) the series (Σa_n ^2 ) and Σa_(n+1) ^2 cv ⇒ Σa_n a_(n+1) converges](https://www.tinkutara.com/question/Q134080.png)
$$\Sigma\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} \leqslant\left(\Sigma\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\Sigma\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{cauchy}\:\mathrm{shwarz}\right)\:\:\mathrm{the}\:\mathrm{series}\:\left(\Sigma\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \right)\:\mathrm{and}\:\Sigma\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} \:\mathrm{cv}\:\Rightarrow \\ $$$$\Sigma\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} \mathrm{converges} \\ $$