Question

Regarding equivalent definition of Essential singularity.

score 43 · Answer 1 · 2026-04-09 05:35:56

43

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Regarding equivalent definition of Essential singularity.

Published on 09 Apr 2026 - 5:35

score 74 · Answer 2 · 2026-04-22 06:59:20

74

Views

Interval of convergence of the infinite series $g(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{1+x^{2n}}$

Published on 22 Apr 2026 - 6:59

score 82 · Answer 3 · 2026-04-17 01:18:59

82

Views

prove $\int_0^1 \frac{x^\sqrt{2}}{1+x} dx= \frac{1}{\sqrt{2} + 1} - \frac{1}{\sqrt{2}+2} + \frac{1}{\sqrt{2}+3}-\frac{1}{\sqrt{2}+4}+\ldots$

Published on 17 Apr 2026 - 1:18

score 101 · Answer 4 · 2026-04-21 16:22:25

101

Views

How do we show that $\sin{x}$ is in fact equal to the infinite series $\sum\limits_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$?

Published on 21 Apr 2026 - 16:22

score 26 · Answer 5 · 2026-04-14 20:12:36

26

Views

Change of order of integration with the inner integral being improper and uniform convergent on the integration segment of the outer integral

Published on 14 Apr 2026 - 20:12

score 51 · Answer 6 · 2026-04-10 02:23:13

51

Views

What actually is needed in proving a function is continuous at a point x0

Published on 10 Apr 2026 - 2:23

score 34 · Answer 7 · 2026-04-17 13:54:38

34

Views

Show that each element of $x_n^{\prime}A_n$ is $o(1)$

Published on 17 Apr 2026 - 13:54

score 43 · Answer 8 · 2026-04-16 10:59:05

43

Views

Prove that the series $f(x) = \sum_{n=1}^{\infty}2^n \sin(\frac{1}{3^nx})$ defines a continuous function on $(0, +\infty)$

Published on 16 Apr 2026 - 10:59

score 52 · Answer 9 · 2026-04-13 23:23:28

52

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Can the forward direction of Arzela-Ascoli follow from Heine-Borel on $f(k/2^n)$?

Published on 13 Apr 2026 - 23:23

score 36 · Answer 10 · 2026-04-13 18:23:22

36

Views

Finding a suitable mollifier $\phi$ so that $\lim_{k\to\infty}||f - h\ast\phi||_\infty=0$ for a continuous function $f$ vanishing at the infinity

Published on 13 Apr 2026 - 18:23

score 133 · Answer 11 · 2026-04-22 06:59:19

133

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Convergence, continuity and differentiability of $f(x)=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+2}-\frac{1}{x+3}+\ldots$

Published on 22 Apr 2026 - 6:59

score 116 · Answer 12 · 2026-04-21 16:22:24

116

Views

If $f_n \to f$ and for every $n > 0$ $f_n$ is uniformly continuous then f is uniformly continuous. Where did my proof fail?

Published on 21 Apr 2026 - 16:22

score 122 · Answer 13 · 2026-04-22 06:56:38

122

Views

Exploring the continuous nowhere differentiable function $g(x) = \sum_{n=0}^{\infty} \frac{\cos {2^n x}}{2^n}$

Published on 22 Apr 2026 - 6:56

score 298 · Answer 14 · 2026-04-22 06:56:56

298

Views

Is $f(x)=\sum_{k=1}^{\infty} \frac{\sin (x/k)}{k}$ continuous, differentiable and twice-differentiable?

Published on 22 Apr 2026 - 6:56

score 68 · Answer 15 · 2026-04-17 04:16:42

68

Views

Let $(X,d)$ be a metric space, and $f_n: X \to \mathbb{R}$ s.t $f_n \to f$ uniformly, f continuous. Show that if $x_n \to x$ then $f_n(x_n) \to f(x)$

Published on 17 Apr 2026 - 4:16

List Question

Regarding equivalent definition of Essential singularity.

Interval of convergence of the infinite series $g(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{1+x^{2n}}$

prove $\int_0^1 \frac{x^\sqrt{2}}{1+x} dx= \frac{1}{\sqrt{2} + 1} - \frac{1}{\sqrt{2}+2} + \frac{1}{\sqrt{2}+3}-\frac{1}{\sqrt{2}+4}+\ldots$

How do we show that $\sin{x}$ is in fact equal to the infinite series $\sum\limits_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$?

Change of order of integration with the inner integral being improper and uniform convergent on the integration segment of the outer integral

What actually is needed in proving a function is continuous at a point x0

Show that each element of $x_n^{\prime}A_n$ is $o(1)$

Prove that the series $f(x) = \sum_{n=1}^{\infty}2^n \sin(\frac{1}{3^nx})$ defines a continuous function on $(0, +\infty)$

Can the forward direction of Arzela-Ascoli follow from Heine-Borel on $f(k/2^n)$?

Finding a suitable mollifier $\phi$ so that $\lim_{k\to\infty}||f - h\ast\phi||_\infty=0$ for a continuous function $f$ vanishing at the infinity

Convergence, continuity and differentiability of $f(x)=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+2}-\frac{1}{x+3}+\ldots$

If $f_n \to f$ and for every $n > 0$ $f_n$ is uniformly continuous then f is uniformly continuous. Where did my proof fail?

Exploring the continuous nowhere differentiable function $g(x) = \sum_{n=0}^{\infty} \frac{\cos {2^n x}}{2^n}$

Is $f(x)=\sum_{k=1}^{\infty} \frac{\sin (x/k)}{k}$ continuous, differentiable and twice-differentiable?

Let $(X,d)$ be a metric space, and $f_n: X \to \mathbb{R}$ s.t $f_n \to f$ uniformly, f continuous. Show that if $x_n \to x$ then $f_n(x_n) \to f(x)$

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