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

$$\mathrm{If}\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}}\mathrm{is}\mathrm{a}\mathrm{convergent}\mathrm{series}\mathrm{of}$$$$\mathrm{nonnegative}\mathrm{terms},\mathrm{what}\mathrm{can}\mathrm{be}$$$$\mathrm{said}\mathrm{about}\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}}.{\mathrm{a}}_{\mathrm{n}+1}?$$$$\left(\mathrm{a}\right)\mathrm{always}\mathrm{converges}$$$$\left(\mathrm{b}\right)\mathrm{always}\mathrm{diverges}$$$$\left(\mathrm{c}\right)\mathrm{may}\mathrm{converges}\mathrm{or}\mathrm{diverge}$$

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

$$\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}}{\mathrm{a}}_{\mathrm{n}+1}⩽{\left(\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}}^2\right)}^{\frac{1}{2}}{\left(\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}+1}^2\right)}^{\frac{1}{2}}\left(\mathrm{cauchy}\mathrm{shwarz}\right)\mathrm{the}\mathrm{series}\left(\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}}^2\right)\mathrm{and}\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}+1}^2\mathrm{cv}\Rightarrow$$$$\mathrm{\Sigma }{\mathrm{a}}_{\mathrm{n}}{\mathrm{a}}_{\mathrm{n}+1}\mathrm{converges}$$

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