Question

Let $\sum_{k=1}^\infty a_n$ be convergent show that $\sum_{k=1}^\infty n(a_n-a_{n+1})$ converges

score 489 · Answer 1 · 2018-12-25 12:34:02

489

Views

Let $\sum_{k=1}^\infty a_n$ be convergent show that $\sum_{k=1}^\infty n(a_n-a_{n+1})$ converges

Published on 25 Dec 2018 - 12:34

score 1k · Answer 2 · 2018-12-25 20:28:55

1k

Views

How can I prove that $x_{n+1} = x_n \sin(x_n)$ converges for any $x_0$?

Published on 25 Dec 2018 - 20:28

score 99 · Answer 3 · 2026-03-29 15:15:17

99

Views

Convergence in $(L_P(a,b),\|.\|_p)$ doesn't imply convergence in $(C(a,b),\sup)$

Published on 29 Mar 2026 - 15:15

score 866 · Answer 4 · 2018-12-26 13:10:38

866

Views

$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$ converges or diverges?

Published on 26 Dec 2018 - 13:10

score 565 · Answer 5 · 2026-03-27 23:35:29

565

Views

An example of a divergent sequence $\{x_n\}$ but with $\lim_{n\to\infty} |x_{n+p} - x_n| = 0$ for any $p \in \Bbb N$

Published on 27 Mar 2026 - 23:35

score 65 · Answer 6 · 2026-03-29 18:33:36

65

Views

Is there a power series $\sum_{n=1}^{\infty}a_nx^n$ example such that $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ divergent?

Published on 29 Mar 2026 - 18:33

score 168 · Answer 7 · 2018-12-26 15:53:59

168

Views

Proof verification. Show that if $\{x_n\}$ diverges then there must be a sequence $\{p_n\}\subset\Bbb N$ such that $\lim(x_{n+p_n} - x_n)\ne 0$

Published on 26 Dec 2018 - 15:53

score 87 · Answer 8 · 2026-04-08 22:22:13

87

Views

How to use infinite series to bound $\pi$.

Published on 08 Apr 2026 - 22:22

score 135 · Answer 9 · 2018-12-26 22:44:03

135

Views

Characterisation of sequences such that every bounded subsequence converges

Published on 26 Dec 2018 - 22:44

score 128 · Answer 10 · 2026-03-27 23:46:42

128

Views

If $\sum_{n=1}^{\infty} a_n$ converges, does $\sum_{n=1}^{\infty}\frac{1}{a_n}$ diverge?

Published on 27 Mar 2026 - 23:46

score 157 · Answer 11 · 2018-12-27 17:00:30

157

Views

What is the Convergence Radius and Domain of $\sum\limits_{n=0}^\infty \frac{1}{2^{n}} z^{2^n}$?

Published on 27 Dec 2018 - 17:00

score 102 · Answer 12 · 2026-03-26 06:34:53

102

Views

A Closed form for the $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3}$

Published on 26 Mar 2026 - 6:34

score 39 · Answer 13 · 2026-04-11 11:14:20

39

Views

In functional analysis, is there a commonly accepted short-hand notation for specific types of convergence?

Published on 11 Apr 2026 - 11:14

score 69 · Answer 14 · 2026-03-26 02:54:04

69

Views

Meaning of being a Cauchy sequence

Published on 26 Mar 2026 - 2:54

score 74 · Answer 15 · 2026-04-11 13:15:25

74

Views

$g_n = \max \{\min (f_n, g), -g\} \to f$

Published on 11 Apr 2026 - 13:15

List Question

Let $\sum_{k=1}^\infty a_n$ be convergent show that $\sum_{k=1}^\infty n(a_n-a_{n+1})$ converges

How can I prove that $x_{n+1} = x_n \sin(x_n)$ converges for any $x_0$?

Convergence in $(L_P(a,b),\|.\|_p)$ doesn't imply convergence in $(C(a,b),\sup)$

$\sum_{n=1}^{\infty} \frac{(1/2) + (-1)^{n}}{n}$ converges or diverges?

An example of a divergent sequence $\{x_n\}$ but with $\lim_{n\to\infty} |x_{n+p} - x_n| = 0$ for any $p \in \Bbb N$

Is there a power series $\sum_{n=1}^{\infty}a_nx^n$ example such that $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ divergent?

Proof verification. Show that if $\{x_n\}$ diverges then there must be a sequence $\{p_n\}\subset\Bbb N$ such that $\lim(x_{n+p_n} - x_n)\ne 0$

How to use infinite series to bound $\pi$.

Characterisation of sequences such that every bounded subsequence converges

If $\sum_{n=1}^{\infty} a_n$ converges, does $\sum_{n=1}^{\infty}\frac{1}{a_n}$ diverge?

What is the Convergence Radius and Domain of $\sum\limits_{n=0}^\infty \frac{1}{2^{n}} z^{2^n}$?

A Closed form for the $\sum_{n=1}^{\infty}\sum_{k=1}^{n}\frac{1}{(25k^2+25k+4)(n-k+1)^3}$

In functional analysis, is there a commonly accepted short-hand notation for specific types of convergence?

Meaning of being a Cauchy sequence

$g_n = \max \{\min (f_n, g), -g\} \to f$

Trending Questions

Popular # Hahtags

Popular Questions