Question

Show that $\sum_{k=1}^{\infty}\frac{2^{-k}}{k}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}$ without evaluating either sum

score 149 · Answer 1 · 2026-04-12 02:08:01

149

Views

Show that $\sum_{k=1}^{\infty}\frac{2^{-k}}{k}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}$ without evaluating either sum

Published on 12 Apr 2026 - 2:08

score 97 · Answer 2 · 2026-04-14 06:08:22

97

Views

Determine whether the following series converges

Published on 14 Apr 2026 - 6:08

score 88 · Answer 3 · 2026-04-16 03:51:35

88

Views

What is known about sequences $a_{n}$ such that $\sum_{n=1}^{N}(-1)^{a_{n}}n = 0$?

Published on 16 Apr 2026 - 3:51

score 121 · Answer 4 · 2026-04-13 20:27:02

121

Views

Convergence of sum of $\sum_{n=1}^\infty \frac{\ln n}{n^{2p}}$

Published on 13 Apr 2026 - 20:27

score 48 · Answer 5 · 2026-04-24 08:14:57

48

Views

Determine if this series : $\sum_{n=1}^\infty \frac{2n-1}{(\sqrt2)^n}$ converges or diverges.

Published on 24 Apr 2026 - 8:14

score 180 · Answer 6 · 2026-04-12 03:28:16

180

Views

Is there a way to sum $\sum_{n=1}^{\infty}\left[ \frac{y}{n} - \tan^{-1}\left( \frac{y}{n} \right)\right]$

Published on 12 Apr 2026 - 3:28

score 116 · Answer 7 · 2026-04-10 16:15:16

116

Views

Real Analysis Sequence limit

Published on 10 Apr 2026 - 16:15

score 136 · Answer 8 · 2026-04-16 17:41:43

136

Views

Why am I getting a negative sum? $\frac{1}{1\cdot2} - \frac{1}{2\cdot3}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+ \cdots$

Published on 16 Apr 2026 - 17:41

score 98 · Answer 9 · 2026-04-13 00:54:14

98

Views

How do I calculate the sum of $\sum_{n=0}^{\infty}{x^{n}\over n!}\cdot{{x^2\over n+2}}?$

Published on 13 Apr 2026 - 0:54

score 221 · Answer 10 · 2026-04-10 06:48:57

221

Views

What does "monotone functions are converging to a continuous limit" mean?

Published on 10 Apr 2026 - 6:48

score 90 · Answer 11 · 2026-04-15 13:02:25

90

Views

Express $\int_0^a x^2\sin(x^3)dx$ as power series

Published on 15 Apr 2026 - 13:02

score 1.3k · Answer 12 · 2026-04-17 23:15:55

1.3k

Views

Sum of reciprocals of odd numbers that add up to $2$

Published on 17 Apr 2026 - 23:15

score 1.5k · Answer 13 · 2026-04-13 00:16:37

1.5k

Views

Taylor series expansion of the cotangent function and denominators of the Riemann zeta function at positive even integers

Published on 13 Apr 2026 - 0:16

score 406 · Answer 14 · 2026-04-14 23:28:25

406

Views

Prove that every positive rational number can be written in factorial base

Published on 14 Apr 2026 - 23:28

score 7.8k · Answer 15 · 2026-04-16 06:11:12

7.8k

Views

Third degree Taylor series of $f(x) = e^x \cos{x} $

Published on 16 Apr 2026 - 6:11

List Question

Show that $\sum_{k=1}^{\infty}\frac{2^{-k}}{k}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}$ without evaluating either sum

Determine whether the following series converges

What is known about sequences $a_{n}$ such that $\sum_{n=1}^{N}(-1)^{a_{n}}n = 0$?

Convergence of sum of $\sum_{n=1}^\infty \frac{\ln n}{n^{2p}}$

Determine if this series : $\sum_{n=1}^\infty \frac{2n-1}{(\sqrt2)^n}$ converges or diverges.

Is there a way to sum $\sum_{n=1}^{\infty}\left[ \frac{y}{n} - \tan^{-1}\left( \frac{y}{n} \right)\right]$

Real Analysis Sequence limit

Why am I getting a negative sum? $\frac{1}{1\cdot2} - \frac{1}{2\cdot3}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+ \cdots$

How do I calculate the sum of $\sum_{n=0}^{\infty}{x^{n}\over n!}\cdot{{x^2\over n+2}}?$

What does "monotone functions are converging to a continuous limit" mean?

Express $\int_0^a x^2\sin(x^3)dx$ as power series

Sum of reciprocals of odd numbers that add up to $2$

Taylor series expansion of the cotangent function and denominators of the Riemann zeta function at positive even integers

Prove that every positive rational number can be written in factorial base

Third degree Taylor series of $f(x) = e^x \cos{x} $

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