Question

Does $\sum^{\infty}_{n=1} \frac{1}{a_n+1}$ with $a_{n+1}=a_n(a_n+1)$ and $a_1=\frac12$ have a closed form?

score 236 · Answer 1 · 2021-03-28 22:37:25

236

Views

Does $\sum^{\infty}_{n=1} \frac{1}{a_n+1}$ with $a_{n+1}=a_n(a_n+1)$ and $a_1=\frac12$ have a closed form?

Published on 28 Mar 2021 - 22:37

score 209 · Answer 2 · 2021-04-10 22:35:55

209

Views

Prove that $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{F_kF_{k+1}}=\frac{1}{\phi}$.

Published on 10 Apr 2021 - 22:35

score 139 · Answer 3 · 2021-04-11 10:35:15

139

Views

Visual proof of $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + ... = 1$

Published on 11 Apr 2021 - 10:35

score 59 · Answer 4 · 2021-04-16 18:16:51

59

Views

Prove the following series converges and has the sum indicated using telescoping: $\sum^{\infty}_{n=1}\frac{2}{3^{n-1}}=3$

Published on 16 Apr 2021 - 18:16

score 116 · Answer 5 · 2021-04-18 19:16:12

116

Views

Compute using the telescoping series $\sum_{n=1}^{\infty} \frac{n}{(n+1)(n+2)(n+3)} = \frac{1}{4}$

Published on 18 Apr 2021 - 19:16

score 60 · Answer 6 · 2021-04-30 19:48:38

60

Views

Find $\sum_{i=1}^n \frac{\sqrt{i+1}-\sqrt{i}}{\sqrt{i^2+i}}$

Published on 30 Apr 2021 - 19:48

score 47 · Answer 7 · 2021-05-01 17:28:19

47

Views

How to compute the limit of the infinite series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)\ldots(n+m)}$, $m \in \mathbf{N}$

Published on 01 May 2021 - 17:28

score 49 · Answer 8 · 2021-05-02 11:57:00

49

Views

Simplification of a telescopic series

Published on 02 May 2021 - 11:57

score 80 · Answer 9 · 2021-05-15 10:42:35

80

Views

Prove that, $\sum_{k=1}^{n-1} \frac{1}{\sqrt{a_k} + \sqrt{a_{k+1}}} = \frac{n-1}{\sqrt{a_1} + \sqrt{a_n}}.$

Published on 15 May 2021 - 10:42

score 169 · Answer 10 · 2021-05-27 12:45:27

169

Views

Summation of powers of r and reciprocal of binomial coefficient

Published on 27 May 2021 - 12:45

score 286 · Answer 11 · 2021-06-01 20:58:40

286

Views

How to find if a series is telescopic

Published on 01 Jun 2021 - 20:58

score 104 · Answer 12 · 2021-06-02 10:08:03

104

Views

Summation of a finite sequence

Published on 02 Jun 2021 - 10:08

score 65 · Answer 13 · 2021-06-05 16:02:50

65

Views

How to evaluate the following Product?

Published on 05 Jun 2021 - 16:02

score 142 · Answer 14 · 2021-09-04 05:03:48

142

Views

Value of $\sum_{n=0}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$ $?$

Published on 04 Sep 2021 - 5:03

score 447 · Answer 15 · 2021-11-08 14:23:45

447

Views

Proving $\sum_{n=1}^{99}\frac{\sqrt{n+1}-\sqrt{n}}{2n+1}\lt\frac9{20}$

Published on 08 Nov 2021 - 14:23

List Question

Does $\sum^{\infty}_{n=1} \frac{1}{a_n+1}$ with $a_{n+1}=a_n(a_n+1)$ and $a_1=\frac12$ have a closed form?

Prove that $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{F_kF_{k+1}}=\frac{1}{\phi}$.

Visual proof of $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + ... = 1$

Prove the following series converges and has the sum indicated using telescoping: $\sum^{\infty}_{n=1}\frac{2}{3^{n-1}}=3$

Compute using the telescoping series $\sum_{n=1}^{\infty} \frac{n}{(n+1)(n+2)(n+3)} = \frac{1}{4}$

Find $\sum_{i=1}^n \frac{\sqrt{i+1}-\sqrt{i}}{\sqrt{i^2+i}}$

How to compute the limit of the infinite series $\sum_{n=1}^{\infty} \frac{1}{n(n+1)\ldots(n+m)}$, $m \in \mathbf{N}$

Simplification of a telescopic series

Prove that, $\sum_{k=1}^{n-1} \frac{1}{\sqrt{a_k} + \sqrt{a_{k+1}}} = \frac{n-1}{\sqrt{a_1} + \sqrt{a_n}}.$

Summation of powers of r and reciprocal of binomial coefficient

How to find if a series is telescopic

Summation of a finite sequence

How to evaluate the following Product?

Value of $\sum_{n=0}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$ $?$

Proving $\sum_{n=1}^{99}\frac{\sqrt{n+1}-\sqrt{n}}{2n+1}\lt\frac9{20}$

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