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if $\limsup a_{n+1}/a_n = L < 1$, then $\sum a_n$ converges

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#sequences-and-series #limits #limsup-and-liminf
5.2k
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Showing that two definitions of $\limsup$ are equivalent

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#real-analysis #definition #limsup-and-liminf
603
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Is there a geometrical meaning of $\limsup_{n\to\infty }A_n$?

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#measure-theory #elementary-set-theory #limsup-and-liminf
368
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Twin prime conjecture implies $\limsup_{n\to\infty}\frac{\sigma(n)\pi(n)}{n^2}\left(\pi(\log n)-\frac{\pi_2(\log n)}{2C_2}\right)=e^{\gamma}$?

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#proof-verification #asymptotics #analytic-number-theory #limsup-and-liminf #conjectures
125
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What does it mean to be less than a constant + $o(1)$?

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#asymptotics #definition #limsup-and-liminf
135
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Proof of $ \limsup\limits_{n\rightarrow\infty} a_n b_n \le (\limsup\limits_{n\rightarrow\infty} a_n)(\limsup\limits_{n\rightarrow\infty} b_n)$

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#real-analysis #limsup-and-liminf
141
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Using measures on symmetric differences

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#real-analysis #measure-theory #lebesgue-measure #limsup-and-liminf
61
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Let $A$ be a bounded sequence in $\mathbb{R}$ and $c > \lim\sup A$. Show that only finitely many terms of $A$ are greater than or equal to $c$.

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#real-analysis #limsup-and-liminf
787
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Is it correct to say that ($\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) \supseteq \limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}$?

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#sequences-and-series #probability-theory #random-variables #limsup-and-liminf #borel-cantelli-lemmas
42
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Certain limit implies Cauchy sequence

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#calculus #analysis #limsup-and-liminf
118
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Find $P(\limsup A_n)$ given $P(A_{n+1}) \ge \frac{n}{n+1} P(A_n)$

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#probability-theory #measure-theory #limsup-and-liminf #independence #borel-cantelli-lemmas
206
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$\liminf |2^m - 3^n|$

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#number-theory #limsup-and-liminf
46
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Find a sequence of RV s.t. $X_n \to 0$ in prob. but its Variance does not?

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#probability-theory #convergence-divergence #limsup-and-liminf #probability-limit-theorems
68
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Limit laws how to write an integral as a max of a sum

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#exponential-function #limsup-and-liminf
82
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Existence of limits of these two sequence.

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#sequences-and-series #divergent-series #limsup-and-liminf #cauchy-sequences

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