Let $\sum_{n=1}^\infty a_{n}$ for $a_{n}\gt 0$. Show that $\lim_{t\to \infty}\frac1{t}\sum_{n=1}^\infty\lfloor ta_{n}\rfloor =\sum_{n=1}^\infty a_{n}$

score 120 · Answer 11 · 2026-04-14 02:55:34

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Show that $\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$

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score 60 · Answer 12 · 2026-04-11 02:18:04

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convergence of series : rational with exponentials

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score 211 · Answer 13 · 2026-04-17 06:21:00

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Evaluating $\frac{1^2}{2^1}+\frac{3^2}{2^2}+\frac{5^2}{2^3}+\cdots$ using sigma notation

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score 1k · Answer 14 · 2026-04-25 16:17:10

Define a sequence $(a_n)$ as follows $a_1 = 1$ and for $n\geq 1,\;a_{n+1} = (-1)^n\frac{1}{2} \left(|a_n| + \frac {2}{|a_n|}\right)$

score 181 · Answer 15 · 2026-04-15 12:05:23

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Figuring out the nth derivative of a Maclaurin series

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Let $\sum_{n=1}^\infty a_{n}$ for $a_{n}\gt 0$. Show that $\lim_{t\to \infty}\frac1{t}\sum_{n=1}^\infty\lfloor ta_{n}\rfloor =\sum_{n=1}^\infty a_{n}$

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