Linked Questions

61
votes
12answers
5k views

The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

What is the sum of the 'second half' of the harmonic series? $$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$ More precisely, what is the limit of the above sequence of partial ...
92
votes
7answers
12k views

Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...
19
votes
5answers
1k views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: $$\...
13
votes
7answers
28k views

Summing Finitely Many Terms of Harmonic Series: $\sum_{k=a}^{b} \frac{1}{k}$

How do I calculate sum of a finite harmonic series of the form : $$\sum_{k=a}^{b} \frac{1}{k} = \frac{1}{a} + \frac{1}{a+1} + \frac{1}{a+2} +\cdots \frac{1}{b}.$$ Is there a general formula for this?...
14
votes
4answers
721 views

Series for logarithms

This is more of a challenge than a question, but I thought I'd share anyway. Prove the following identities, and prove that the pattern continues. \begin{equation*} \sum_{n=0}^\infty\left(\frac{1}{2n+...
5
votes
5answers
692 views

Can we assign a value to the sum of the reciprocals of the natural numbers?

I know the sum of the reciprocals of the natural numbers diverges to infinity, but I want to know what value can be assigned to it. $$\sum_{n=1}^{\infty}\frac1n=\frac11+\frac12+\frac13+\frac14+\dots=...
5
votes
3answers
387 views

An algorithm for making conditionally convergent series take arbitrary values?

This thread reminded me of an old unsettled question I have. Given an arbitrary conditionally convergent series $\beta=\sum\limits_{k=1}^\infty a_k$ and a target value $\alpha$, is there an algorithm ...
5
votes
3answers
2k views

Show $\sum_\limits{k=1}^{\infty}1/k$ does not converge. [duplicate]

Show $$\sum_\limits{k=1}^\infty \frac 1 k$$ does not converge. Attempt: Let $s_n=\sum_\limits{k=1}^{n}1/k$, and let $\epsilon=1/2$. For all $N\in\mathbb{N}$, we have $$\left|s_{2n}-s_n\right|=\left|\...
1
vote
3answers
718 views

Difference of harmonic series

Proving convergence: $$\sum_{n=1}^\infty (-1)^{n-1}\frac1n$$ Just wanted to confirm if the reason they converge is due to the fact that for n= 1, 3, 5, ... we have a positive harmonic series and for ...
0
votes
2answers
304 views

Find the sum of the series (rearranged harmonic series)

Find the value of the series: $$1-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}-\frac{1}{8}+\frac{1}{3}-\frac{1}{10}-\frac{1}{12}-\frac{1}{14}-\frac{1}{16}+\frac{1}{5}\cdots$$ I know that the alternated ...