Question

Calculate the sum $\sum_{k=0}^{\infty} a^{\frac{k(k-1)}{2}}r^k$

score 49 · Answer 1 · 2026-03-26 23:11:09

49

Views

Calculate the sum $\sum_{k=0}^{\infty} a^{\frac{k(k-1)}{2}}r^k$

Published on 26 Mar 2026 - 23:11

score 135 · Answer 2 · 2026-04-15 09:27:48

135

Views

Sum of the product of binomial coefficients

Published on 15 Apr 2026 - 9:27

score 86 · Answer 3 · 2026-04-09 12:57:59

86

Views

Analytical Formula for Adding up Rectangular Pyramid Block Numbers

Published on 09 Apr 2026 - 12:57

score 26 · Answer 4 · 2026-04-16 14:14:20

26

Views

I'm trying to determine a big O notation for the following - just wanted to confirm if this method makes sense?

Published on 16 Apr 2026 - 14:14

score 53 · Answer 5 · 2026-04-11 21:42:37

53

Views

Is there a closed form to this summation?

Published on 11 Apr 2026 - 21:42

score 120 · Answer 6 · 2026-04-19 07:31:39

120

Views

Simplify $\sum_{k = 0}^{[p/3] - 2} \sum_{i = 1}^{[p/(k + 1)] - 3} \sum_{j = 0}^{p - (k + 1) (i + 2) - 1} {\delta}_{N, i\, p + j + k + 1}$

Published on 19 Apr 2026 - 7:31

score 240 · Answer 7 · 2026-03-25 17:41:55

240

Views

Let $a, b, c, d \in R^+$ such that $a + b + c + d = 1$. Prove that $\frac{a^3}{b+c}+\frac{b^3}{c+d}+\frac{c^3}{d+a}+\frac{d^3}{a+b} \geq \frac{1}{8}$

Published on 25 Mar 2026 - 17:41

score 72 · Answer 8 · 2026-05-10 16:13:41

72

Views

Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$

Published on 10 May 2026 - 16:13

score 783 · Answer 9 · 2026-05-10 23:31:39

783

Views

Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

Published on 10 May 2026 - 23:31

score 190 · Answer 10 · 2026-04-17 14:26:34

190

Views

Combinations series: $\frac{{n \choose 1}(n-1)^3+{n \choose 3}(n-3)^3+\ldots}{n^2(n+3)\cdot 2^n}$

Published on 17 Apr 2026 - 14:26

score 44 · Answer 11 · 2026-04-24 22:30:38

44

Views

What's a good symbol to represent many summations? Can a tensor product glyph be used?

Published on 24 Apr 2026 - 22:30

score 80 · Answer 12 · 2026-04-12 03:08:43

80

Views

Identity involving Binomial terms arising from number of paths in a graph with $h$ hops.

Published on 12 Apr 2026 - 3:08

score 168 · Answer 13 · 2026-04-11 07:46:10

168

Views

Closed form for Lobb Numbers

Published on 11 Apr 2026 - 7:46

score 676 · Answer 14 · 2026-05-11 03:13:49

676

Views

Sum of discrete Gaussian series

Published on 11 May 2026 - 3:13

score 203 · Answer 15 · 2026-04-14 09:41:22

203

Views

Closed form expression for the expected Euclidean distance between $2$ randomly chosen lattice points among $N$ points in a rectangular region

Published on 14 Apr 2026 - 9:41

List Question

Calculate the sum $\sum_{k=0}^{\infty} a^{\frac{k(k-1)}{2}}r^k$

Sum of the product of binomial coefficients

Analytical Formula for Adding up Rectangular Pyramid Block Numbers

I'm trying to determine a big O notation for the following - just wanted to confirm if this method makes sense?

Is there a closed form to this summation?

Simplify $\sum_{k = 0}^{[p/3] - 2} \sum_{i = 1}^{[p/(k + 1)] - 3} \sum_{j = 0}^{p - (k + 1) (i + 2) - 1} {\delta}_{N, i\, p + j + k + 1}$

Let $a, b, c, d \in R^+$ such that $a + b + c + d = 1$. Prove that $\frac{a^3}{b+c}+\frac{b^3}{c+d}+\frac{c^3}{d+a}+\frac{d^3}{a+b} \geq \frac{1}{8}$

Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$

Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

Combinations series: $\frac{{n \choose 1}(n-1)^3+{n \choose 3}(n-3)^3+\ldots}{n^2(n+3)\cdot 2^n}$

What's a good symbol to represent many summations? Can a tensor product glyph be used?

Identity involving Binomial terms arising from number of paths in a graph with $h$ hops.

Closed form for Lobb Numbers

Sum of discrete Gaussian series

Closed form expression for the expected Euclidean distance between $2$ randomly chosen lattice points among $N$ points in a rectangular region

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