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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
$$\mathrm{If}\:\Sigma\:\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{series}\:\mathrm{of} \\ $$$$\mathrm{nonnegative}\:\mathrm{terms},\mathrm{what}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{said}\:\mathrm{about}\:\Sigma\:\mathrm{a}_{\mathrm{n}} .\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{always}\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{always}\:\mathrm{diverges} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{may}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverge} \\ $$
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
$$\Sigma\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} \leqslant\left(\Sigma\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\Sigma\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{cauchy}\:\mathrm{shwarz}\right)\:\:\mathrm{the}\:\mathrm{series}\:\left(\Sigma\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \right)\:\mathrm{and}\:\Sigma\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} \:\mathrm{cv}\:\Rightarrow \\ $$$$\Sigma\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} \mathrm{converges} \\ $$

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