Question

Why does $\sum_{i=1}^x \frac xi - \lfloor \frac xi \rceil \approx 0.03648x$ instead of $x(1-2\gamma)$

score 110 · Answer 1 · 2026-03-26 17:14:35

110

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Why does $\sum_{i=1}^x \frac xi - \lfloor \frac xi \rceil \approx 0.03648x$ instead of $x(1-2\gamma)$

Published on 26 Mar 2026 - 17:14

score 67 · Answer 2 · 2017-09-06 02:13:22

67

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How do you calculate the generic formula for $ \sum\limits_{i=0}^{k} \frac 1 {(n-i)}$

Published on 06 Sep 2017 - 2:13

score 2.3k · Answer 3 · 2017-09-06 15:30:35

2.3k

Views

Sum of powers of Harmonic Numbers

Published on 06 Sep 2017 - 15:30

score 290 · Answer 4 · 2017-09-08 17:04:24

290

Views

Existence of a sequence $\{\epsilon_n\}_{n\ge 1}$ such that $\sum\limits_{n=1}^{\infty}\frac{1}{n^{\varepsilon_n}} $ converges

Published on 08 Sep 2017 - 17:04

score 511 · Answer 5 · 2017-09-09 14:30:15

511

Views

Is there a closed form for $\sum _{k=1}^n \frac{1}{k}H_{k-1}^2$?

Published on 09 Sep 2017 - 14:30

score 238 · Answer 6 · 2017-09-13 08:07:12

238

Views

Fibonacci and Harmonic

Published on 13 Sep 2017 - 8:07

score 1.1k · Answer 7 · 2017-09-14 09:36:46

1.1k

Views

Is there a closed form for the alternating series of inverse harmonic numbers?

Published on 14 Sep 2017 - 9:36

score 441 · Answer 8 · 2017-09-16 16:23:03

441

Views

How to compute $S_{2016}=\sum\limits_{k=1}^{2016}\left(\sum\limits_{n=k}^{2016}\frac1n\right)^2+\sum\limits_{k=1}^{2016}\frac1k$?

Published on 16 Sep 2017 - 16:23

score 705 · Answer 9 · 2026-03-26 09:26:52

705

Views

A Summability methods which sum the harmonic series

Published on 26 Mar 2026 - 9:26

score 63 · Answer 10 · 2026-03-25 20:35:11

63

Views

Partial sum formula for $\sum_{x=1}^n \frac{1}{bx - x^2}$ at specific limit without using digamma function

Published on 25 Mar 2026 - 20:35

score 114 · Answer 11 · 2017-09-27 22:26:38

114

Views

Formulas involving the polynomials $\frac{1}{n!}x^n(a_n-b_nx)^n,$ where the coprime integers $a_n,b_n$ satisfy $H_n=\frac{a_n}{b_n}$ for $n\geq 1$

Published on 27 Sep 2017 - 22:26

score 381 · Answer 12 · 2017-09-29 09:25:29

381

Views

Infinite sum involving complete and incomplete zeta function

Published on 29 Sep 2017 - 9:25

score 164 · Answer 13 · 2017-09-29 13:12:55

164

Views

Exploring the constant $-\int_0^1\log\left(1-H_x\right)\log\left(1+H_{2x}\right)\,dx$, where $H_y$ are harmonic numbers

Published on 29 Sep 2017 - 13:12

score 99 · Answer 14 · 2026-02-24 03:45:16

99

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Is the function $\frac{\operatorname{Harmonic}(x)}{\operatorname{Airy}(x)}$ a convex function for real numbers $0<x<1$?

Published on 24 Feb 2026 - 3:45

score 236 · Answer 15 · 2017-10-03 15:47:18

236

Views

Comparison of sum and integral over Harmonic number

Published on 03 Oct 2017 - 15:47

List Question

Why does $\sum_{i=1}^x \frac xi - \lfloor \frac xi \rceil \approx 0.03648x$ instead of $x(1-2\gamma)$

How do you calculate the generic formula for $ \sum\limits_{i=0}^{k} \frac 1 {(n-i)}$

Sum of powers of Harmonic Numbers

Existence of a sequence $\{\epsilon_n\}_{n\ge 1}$ such that $\sum\limits_{n=1}^{\infty}\frac{1}{n^{\varepsilon_n}} $ converges

Is there a closed form for $\sum _{k=1}^n \frac{1}{k}H_{k-1}^2$?

Fibonacci and Harmonic

Is there a closed form for the alternating series of inverse harmonic numbers?

How to compute $S_{2016}=\sum\limits_{k=1}^{2016}\left(\sum\limits_{n=k}^{2016}\frac1n\right)^2+\sum\limits_{k=1}^{2016}\frac1k$?

A Summability methods which sum the harmonic series

Partial sum formula for $\sum_{x=1}^n \frac{1}{bx - x^2}$ at specific limit without using digamma function

Formulas involving the polynomials $\frac{1}{n!}x^n(a_n-b_nx)^n,$ where the coprime integers $a_n,b_n$ satisfy $H_n=\frac{a_n}{b_n}$ for $n\geq 1$

Infinite sum involving complete and incomplete zeta function

Exploring the constant $-\int_0^1\log\left(1-H_x\right)\log\left(1+H_{2x}\right)\,dx$, where $H_y$ are harmonic numbers

Is the function $\frac{\operatorname{Harmonic}(x)}{\operatorname{Airy}(x)}$ a convex function for real numbers $0<x<1$?

Comparison of sum and integral over Harmonic number

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