Question

Why does $\sum_{n=1}^\infty \frac{\cos x^n}{n}$ converge on $-1\le x \lt 1 $, and uniformly converge on $(-1,0)$

score 117 · Answer 1 · 2026-04-13 23:42:52

117

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Why does $\sum_{n=1}^\infty \frac{\cos x^n}{n}$ converge on $-1\le x \lt 1 $, and uniformly converge on $(-1,0)$

Published on 13 Apr 2026 - 23:42

score 68 · Answer 2 · 2026-04-18 01:08:38

68

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Finding where {$f_n(x)$} converges, where $f_n(x)=x(1-x^n)$

Published on 18 Apr 2026 - 1:08

score 319 · Answer 3 · 2026-03-25 18:08:14

319

Views

Ways to justify this interchange of summation and integration

Published on 25 Mar 2026 - 18:08

score 454 · Answer 4 · 2026-04-14 06:58:21

454

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Convergence stronger than uniform convergence

Published on 14 Apr 2026 - 6:58

score 1.4k · Answer 5 · 2026-04-12 04:34:55

1.4k

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Proof that $\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2}$ convergences uniformly using the Cauchy criterion

Published on 12 Apr 2026 - 4:34

score 239 · Answer 6 · 2026-04-12 18:38:52

239

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Absolute convergence of $\sum\limits_{n=0}^{\infty} \frac {z} {(z+n)^2}$

Published on 12 Apr 2026 - 18:38

score 147 · Answer 7 · 2026-04-15 21:34:02

147

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Show $\frac{\ln{(1+nx^2)}}{2n}$ is not uniform convergent on $[0,\infty)$

Published on 15 Apr 2026 - 21:34

score 22k · Answer 8 · 2026-04-17 09:53:37

22k

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$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Published on 17 Apr 2026 - 9:53

score 199 · Answer 9 · 2026-04-15 19:56:06

199

Views

Prove that $f_n$ converges uniformly on $\mathbb{R}$

Published on 15 Apr 2026 - 19:56

score 330 · Answer 10 · 2026-04-16 22:37:02

330

Views

Point convergence, uniform convergence and near uniform convergence of infinite series $ f_n = x^2 e^{-nx}$

Published on 16 Apr 2026 - 22:37

score 382 · Answer 11 · 2026-04-18 15:03:40

382

Views

Please prove that $\lim\limits_{n \to \infty} \int_0^3 \sqrt{\sin(n/x)+x+1}\,dx$ exists and evaluate it.

Published on 18 Apr 2026 - 15:03

score 237 · Answer 12 · 2026-04-16 08:48:34

237

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Isn't $f_n(x)=\frac{{n^2}\ln x}{x^n}$ with $x\geq1$ uniformly convergent by $T$-test?

Published on 16 Apr 2026 - 8:48

score 302 · Answer 13 · 2026-04-11 16:58:58

302

Views

I need help about some compactness arguments

Published on 11 Apr 2026 - 16:58

score 294 · Answer 14 · 2026-04-16 07:52:47

294

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Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$

Published on 16 Apr 2026 - 7:52

score 3.4k · Answer 15 · 2026-04-16 07:39:06

3.4k

Views

Uniform convergence of the series : $ \sum\frac{1}{n^2 x^2}$, x is not zero.

Published on 16 Apr 2026 - 7:39

List Question

Why does $\sum_{n=1}^\infty \frac{\cos x^n}{n}$ converge on $-1\le x \lt 1 $, and uniformly converge on $(-1,0)$

Finding where {$f_n(x)$} converges, where $f_n(x)=x(1-x^n)$

Ways to justify this interchange of summation and integration

Convergence stronger than uniform convergence

Proof that $\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2}$ convergences uniformly using the Cauchy criterion

Absolute convergence of $\sum\limits_{n=0}^{\infty} \frac {z} {(z+n)^2}$

Show $\frac{\ln{(1+nx^2)}}{2n}$ is not uniform convergent on $[0,\infty)$

$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Prove that $f_n$ converges uniformly on $\mathbb{R}$

Point convergence, uniform convergence and near uniform convergence of infinite series $ f_n = x^2 e^{-nx}$

Please prove that $\lim\limits_{n \to \infty} \int_0^3 \sqrt{\sin(n/x)+x+1}\,dx$ exists and evaluate it.

Isn't $f_n(x)=\frac{{n^2}\ln x}{x^n}$ with $x\geq1$ uniformly convergent by $T$-test?

I need help about some compactness arguments

Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$

Uniform convergence of the series : $ \sum\frac{1}{n^2 x^2}$, x is not zero.

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