Question

Does $\lim_{x\to 1^-}\sum_{n=0}^\infty a_n x^n\neq\infty$ implies $-\infty<\liminf\sum_{n=0}^N a_n\le\limsup\sum_{n=0}^N a_n<+\infty$?

score 152 · Answer 1 · 2026-03-25 04:43:58

152

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Does $\lim_{x\to 1^-}\sum_{n=0}^\infty a_n x^n\neq\infty$ implies $-\infty<\liminf\sum_{n=0}^N a_n\le\limsup\sum_{n=0}^N a_n<+\infty$?

Published on 25 Mar 2026 - 4:43

score 46 · Answer 2 · 2019-08-22 15:17:56

46

Views

Study the convergence of the series

Published on 22 Aug 2019 - 15:17

score 27 · Answer 3 · 2019-08-22 15:54:17

Expansion of $\sum_{n=0}^{\infty} (-1)^n \left(\frac{x^{n+\frac{1}{2}} - x^{-n-\frac{1}{2}}}{x^{\frac{1}{2}} - x^{-\frac{1}{2}}} \right) $

score 88 · Answer 4 · 2019-08-23 11:01:21

88

Views

Why are power series not polynomials?

Published on 23 Aug 2019 - 11:01

score 69 · Answer 5 · 2019-08-23 21:35:27

69

Views

$\sum _{n=1}^{\infty }\:\frac{1}{n}\left(1-z^2\right)^n$ uniform convergence over compacts.

Published on 23 Aug 2019 - 21:35

score 216 · Answer 6 · 2019-08-23 22:12:41

216

Views

Examine convergence of $\sum_{n=1}^\infty \frac{(-1)^n}{n}z^{n(n+1)}$

Published on 23 Aug 2019 - 22:12

score 112 · Answer 7 · 2019-08-24 08:28:46

112

Views

Equivalence of power series

Published on 24 Aug 2019 - 8:28

score 110 · Answer 8 · 2019-08-24 09:13:12

110

Views

Determine radius of convergence of power series

Published on 24 Aug 2019 - 9:13

score 52 · Answer 9 · 2026-04-03 14:03:58

52

Views

How to tell whether a function has a root in a specific set?

Published on 03 Apr 2026 - 14:03

score 121 · Answer 10 · 2019-08-24 21:17:16

121

Views

$ \sum_{k=1}^{\infty}\frac{\log(k)}{k^2(k+1)}$ explicit value.

Published on 24 Aug 2019 - 21:17

score 51 · Answer 11 · 2019-08-25 15:09:37

51

Views

Evaluate these series

Published on 25 Aug 2019 - 15:09

score 69 · Answer 12 · 2019-08-25 19:48:11

69

Views

When limit of complex function exists and nonzero? What did I do wrong?

Published on 25 Aug 2019 - 19:48

score 55 · Answer 13 · 2026-03-25 12:53:51

55

Views

Analytic continuation of a class of functions

Published on 25 Mar 2026 - 12:53

score 91 · Answer 14 · 2019-08-27 14:57:05

91

Views

Upper bound for the sum of $~i^{-1.5}~$

Published on 27 Aug 2019 - 14:57

score 39 · Answer 15 · 2019-08-27 16:11:09

39

Views

Can anybody name this theorem that deals with remainders of power series in connection with radius of convergence?

Published on 27 Aug 2019 - 16:11

List Question

Does $\lim_{x\to 1^-}\sum_{n=0}^\infty a_n x^n\neq\infty$ implies $-\infty<\liminf\sum_{n=0}^N a_n\le\limsup\sum_{n=0}^N a_n<+\infty$?

Study the convergence of the series

Expansion of $\sum_{n=0}^{\infty} (-1)^n \left(\frac{x^{n+\frac{1}{2}} - x^{-n-\frac{1}{2}}}{x^{\frac{1}{2}} - x^{-\frac{1}{2}}} \right) $

Why are power series not polynomials?

$\sum _{n=1}^{\infty }\:\frac{1}{n}\left(1-z^2\right)^n$ uniform convergence over compacts.

Examine convergence of $\sum_{n=1}^\infty \frac{(-1)^n}{n}z^{n(n+1)}$

Equivalence of power series

Determine radius of convergence of power series

How to tell whether a function has a root in a specific set?

$ \sum_{k=1}^{\infty}\frac{\log(k)}{k^2(k+1)}$ explicit value.

Evaluate these series

When limit of complex function exists and nonzero? What did I do wrong?

Analytic continuation of a class of functions

Upper bound for the sum of $~i^{-1.5}~$

Can anybody name this theorem that deals with remainders of power series in connection with radius of convergence?

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