Question

Proof-checking proof of Banach space

score 163 · Answer 1 · 2026-04-13 08:03:00

163

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Proof-checking proof of Banach space

Published on 13 Apr 2026 - 8:03

score 1k · Answer 2 · 2026-04-16 09:57:31

1k

Views

If $\sum a_n$ converges then $\sum a_n^{1-1/n}$ converges.

Published on 16 Apr 2026 - 9:57

score 36 · Answer 3 · 2026-04-08 19:32:11

36

Views

Extending convergence from a point to an interval

Published on 08 Apr 2026 - 19:32

score 321 · Answer 4 · 2026-04-16 18:57:53

321

Views

$\{f_n(x)\}$ sequence of non-negative continuous functions such that $f_{n+1}(x) \le f_n(x)$. Prove $\lim f_n(x) = f(x)$ for all $x \in \mathbb R$.

Published on 16 Apr 2026 - 18:57

score 86 · Answer 5 · 2026-04-16 09:18:02

86

Views

Convergence of $\sum_{n=1}^{\infty} \frac{\frac{1}{2}+(-1)^n}{n}$

Published on 16 Apr 2026 - 9:18

score 94 · Answer 6 · 2026-04-15 20:48:25

Show $\sum_{k=0}^\infty \frac{1}{(k+a)\binom{n+k}{k}}=\frac{-\,n!}{(1-a)_n}\Big(\gamma+\psi(a)+\sum_{k=1}^{n-1}\frac{1}{k}\frac{(1-a)_k}{k!}\Big)$

score 73 · Answer 7 · 2026-05-15 19:30:14

73

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Limit of a sequence using Squeezing Principle: $\sum_{k=1}^n \frac{n}{n^3+k} = \frac{n}{n^3+1}+...+\frac{n}{n^3+n}$.

Published on 15 May 2026 - 19:30

score 2.9k · Answer 8 · 2026-04-16 12:19:13

2.9k

Views

Is it true that if $(a_{n+1}-a_n)\to 0$ and $(a_n)$ is bounded, then $(a_n)$ is convergent

Published on 16 Apr 2026 - 12:19

score 60 · Answer 9 · 2026-05-11 01:19:40

60

Views

How to prove/disprove monotonicity of this function?

Published on 11 May 2026 - 1:19

score 64 · Answer 10 · 2026-05-15 19:31:23

64

Views

Calculus limit with sum of radicals

Published on 15 May 2026 - 19:31

score 1.6k · Answer 11 · 2026-04-10 03:15:09

1.6k

Views

What is an omega sequence?

Published on 10 Apr 2026 - 3:15

score 191 · Answer 12 · 2026-04-16 00:10:22

191

Views

Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $

Published on 16 Apr 2026 - 0:10

score 95 · Answer 13 · 2026-04-15 08:12:17

95

Views

Show that given sequence is convergent.

Published on 15 Apr 2026 - 8:12

score 951 · Answer 14 · 2026-04-16 04:36:44

951

Views

Prove $\sum_{i=0}^n (-1)^{n-i} \binom{n+1}{i} (i+1)^n = (n+2)^n$

Published on 16 Apr 2026 - 4:36

score 68 · Answer 15 · 2026-04-16 10:02:13

68

Views

Then show that $ \ \sqrt[n]{n^3} \to 1 \ \text{as} \ n \to \infty \ $

Published on 16 Apr 2026 - 10:02

List Question

Proof-checking proof of Banach space

If $\sum a_n$ converges then $\sum a_n^{1-1/n}$ converges.

Extending convergence from a point to an interval

$\{f_n(x)\}$ sequence of non-negative continuous functions such that $f_{n+1}(x) \le f_n(x)$. Prove $\lim f_n(x) = f(x)$ for all $x \in \mathbb R$.

Convergence of $\sum_{n=1}^{\infty} \frac{\frac{1}{2}+(-1)^n}{n}$

Show $\sum_{k=0}^\infty \frac{1}{(k+a)\binom{n+k}{k}}=\frac{-\,n!}{(1-a)_n}\Big(\gamma+\psi(a)+\sum_{k=1}^{n-1}\frac{1}{k}\frac{(1-a)_k}{k!}\Big)$

Limit of a sequence using Squeezing Principle: $\sum_{k=1}^n \frac{n}{n^3+k} = \frac{n}{n^3+1}+...+\frac{n}{n^3+n}$.

Is it true that if $(a_{n+1}-a_n)\to 0$ and $(a_n)$ is bounded, then $(a_n)$ is convergent

How to prove/disprove monotonicity of this function?

Calculus limit with sum of radicals

What is an omega sequence?

Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $

Show that given sequence is convergent.

Prove $\sum_{i=0}^n (-1)^{n-i} \binom{n+1}{i} (i+1)^n = (n+2)^n$

Then show that $ \ \sqrt[n]{n^3} \to 1 \ \text{as} \ n \to \infty \ $

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