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Mathematical Analysis, Apostol著, 213页, Exercise 8.24
Given that $∑a_n$ converges, where each $a_n>0$. Prove that $∑(a_na_{n+1})^{1/2}$ also converges. Show that the converse is also true if $\{a_n\}$ is monotonic.
math.stackexchange.com/questions/32546/using-the-partial-summation-formula?noredirect=1
$\sqrt{a_n a_{n+1}} \leq \max\{a_n, a_{n+1}\}\leq a_n + a_{n+1}$.
By comparison test, this implies $\sum \sqrt{ a_n a_{n+1}}$ converges.
Conversely, if $\{a_n\}$ is decreasing and $\sum \sqrt{ a_n a_{n+1}}$, then $\sqrt{a_na_{n+1}}\geq a_{n+1}$. By comparison test, this implies $\sum a_n$ converges. |
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