All Questions
7
questions
0
votes
0
answers
102
views
Is there a formula for $\sum_{i=1}^n \frac 1i$? [duplicate]
Is there a formula for
$$\sum _{i=1}^n\frac{1}{i} \,?$$
Any help would be great
- 17
3
votes
2
answers
144
views
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
As the title says I need to prove the following by induction:
$$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$
When trying to prove that P(n+1) is true if P(n) is, then I ...
- 706
3
votes
0
answers
171
views
How to prove the identity $\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$? [duplicate]
Prove That
$$\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$$
I encountered this identity while reading the article about Harmonic Number on Wikipedia. I thought of using the ...
user313478
1
vote
1
answer
216
views
What is the proof for this sum of sum generalized harmonic number?
I believe this sum: $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}$$ to be equal to $$\frac 12((H_k^{s})^2-H_k^{(2s)})$$ where $H_k^{s}$ is the generalized harmonic number. I only discovered this by ...
- 3,557
3
votes
1
answer
5k
views
Relationship between logarithms and harmonic series
This article on the harmonic series says that
$$\sum_{n=1}^k\,\frac{1}{n} \;=\; \ln k + \gamma + \varepsilon_k$$
where
$$\varepsilon_k\sim\frac{1}{2k}$$
and this seems to generalise to
$$\sum_{n=1}...
- 9,018
41
votes
5
answers
21k
views
How to find the sum of the series $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$?
How to find the sum of the following series?
$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$$
This is a harmonic progression. So, is the following formula correct?
$\frac{(number ~...
- 10k
30
votes
5
answers
13k
views
Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$ [closed]
Is there any formula for this series?
$$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
- 459