Question

Suppose $ \sum a_k x^k $ uniformly converges at $ [0,R) $, then it converges pointwise at $ R $.

score 82 · Answer 1 · 2026-04-16 10:44:13

82

Views

Suppose $ \sum a_k x^k $ uniformly converges at $ [0,R) $, then it converges pointwise at $ R $.

Published on 16 Apr 2026 - 10:44

score 71 · Answer 2 · 2026-04-13 00:56:25

71

Views

Show that if $f \in L^1 \cap C$, then $\sum_{n \in \mathbb{Z}} f(x+n)$ exists and is uniformly continuous

Published on 13 Apr 2026 - 0:56

score 33 · Answer 3 · 2026-04-18 09:54:36

33

Views

Is it true that $ \Im ( \sum_{k=0}^\infty z_k ) = \sum_{k=0}^\infty \Im( z_k)$ and $\Re ( \sum_{k=0}^\infty z_k )= \sum_{k=0}^\infty \Re( z_k) $?

Published on 18 Apr 2026 - 9:54

score 71 · Answer 4 · 2026-04-15 19:58:17

71

Views

Calculate $\lim_{n\rightarrow\infty} \int_{0}^{\pi/2} \frac{dx}{\sum_{k=0}^{n}(\frac{\sin x}{2})^k}$

Published on 15 Apr 2026 - 19:58

score 149 · Answer 5 · 2026-04-16 13:15:55

149

Views

Prove that $\int_{0}^{1} \frac{\ln(x)}{x-1} dx = \sum_{k=1}^{\infty} \frac{1}{k^2}$

Published on 16 Apr 2026 - 13:15

score 63 · Answer 6 · 2026-04-17 04:58:48

63

Views

Why can't we say that $\lim_{n\rightarrow \infty} \int_{1}^{\infty} f_n(x)dx = \int_{1}^{\infty} f(x)dx$ in this case?

Published on 17 Apr 2026 - 4:58

score 49 · Answer 7 · 2026-04-13 10:54:54

49

Views

Runge uniform convergence theorem on closed curves

Published on 13 Apr 2026 - 10:54

score 67 · Answer 8 · 2026-04-16 12:12:02

67

Views

$\int_C|f_n(z)||dz|\leq M$ for each $f_n$. Prove $\{f_n\}$ has a subsequence converging uniformly on compact subsets of $D$.

Published on 16 Apr 2026 - 12:12

score 48 · Answer 9 · 2026-04-16 06:27:05

48

Views

Convergence of this series: $ \sum_{n=1}^\infty \sin\Big(\frac{2x}{x+(n+5)^6}\Big)$

Published on 16 Apr 2026 - 6:27

score 94 · Answer 10 · 2026-04-14 23:31:45

94

Views

Convergence uniformly in $\mathbb{R}$, but not in $L^{2}(\mathbb{R})$, and convergence in $L^{2}(\mathbb{R})$, but not uniformly in $\mathbb{R}$

Published on 14 Apr 2026 - 23:31

score 137 · Answer 11 · 2026-04-15 15:08:24

137

Views

Prove that $\Gamma=\{\gamma:K\longrightarrow (Y, \| . \|) : \gamma\; \text{is continuous and} \; \gamma|_{K_0}=\gamma_0\}$ is a complete metric space.

Published on 15 Apr 2026 - 15:08

score 232 · Answer 12 · 2026-04-15 03:33:35

232

Views

Uniform convergence of a parametric integral. Which solution is correct?

Published on 15 Apr 2026 - 3:33

score 53 · Answer 13 · 2026-04-15 17:13:59

53

Views

Uniform convergence of $\cos(f_n)$.

Published on 15 Apr 2026 - 17:13

score 56 · Answer 14 · 2026-04-16 03:57:02

56

Views

Where can one find good problems like these?

Published on 16 Apr 2026 - 3:57

score 68 · Answer 15 · 2026-04-15 12:20:24

68

Views

Find the value of the given limit:

Published on 15 Apr 2026 - 12:20

List Question

Suppose $ \sum a_k x^k $ uniformly converges at $ [0,R) $, then it converges pointwise at $ R $.

Show that if $f \in L^1 \cap C$, then $\sum_{n \in \mathbb{Z}} f(x+n)$ exists and is uniformly continuous

Is it true that $ \Im ( \sum_{k=0}^\infty z_k ) = \sum_{k=0}^\infty \Im( z_k)$ and $\Re ( \sum_{k=0}^\infty z_k )= \sum_{k=0}^\infty \Re( z_k) $?

Calculate $\lim_{n\rightarrow\infty} \int_{0}^{\pi/2} \frac{dx}{\sum_{k=0}^{n}(\frac{\sin x}{2})^k}$

Prove that $\int_{0}^{1} \frac{\ln(x)}{x-1} dx = \sum_{k=1}^{\infty} \frac{1}{k^2}$

Why can't we say that $\lim_{n\rightarrow \infty} \int_{1}^{\infty} f_n(x)dx = \int_{1}^{\infty} f(x)dx$ in this case?

Runge uniform convergence theorem on closed curves

$\int_C|f_n(z)||dz|\leq M$ for each $f_n$. Prove $\{f_n\}$ has a subsequence converging uniformly on compact subsets of $D$.

Convergence of this series: $ \sum_{n=1}^\infty \sin\Big(\frac{2x}{x+(n+5)^6}\Big)$

Convergence uniformly in $\mathbb{R}$, but not in $L^{2}(\mathbb{R})$, and convergence in $L^{2}(\mathbb{R})$, but not uniformly in $\mathbb{R}$

Prove that $\Gamma=\{\gamma:K\longrightarrow (Y, \| . \|) : \gamma\; \text{is continuous and} \; \gamma|_{K_0}=\gamma_0\}$ is a complete metric space.

Uniform convergence of a parametric integral. Which solution is correct?

Uniform convergence of $\cos(f_n)$.

Where can one find good problems like these?

Find the value of the given limit:

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