Limit of 1/n + ... + 1/2n

Asked Oct 28 '23 at 11:46

Active Oct 28 '23 at 11:46

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Recently my professor gave us the exercise of proving wether the sequence a_n=1/n + 1/n+1 + ... + 1/2n was convergent. It's not very hard to find It's monotonous and 0<=a_n<=1.5 for all n (this has a name, but I don't know how it's spelled in english), then it must converge.

So my question: what does this limit equal? Is it a unique constant difined as the limit, or is it rational, or does it have relation with some other constants?

asked Oct 28 '23 at 11:46

Pablusky

3

You can work this out by using asymptotics for the harmonic numbers. It should be $\ln 2$. It should be possible to prove this directly by showing that $a_n$ approximates the integral $\int_n^{2n} \tfrac 1x ,\mathrm dx$. – Izaak van Dongen Oct 28 '23 at 11:54
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Welcome to MSE. The limit is $\log2$. Besides integral comparison, you can prove it by applying the convexity inequality $\log(1+x)\le x\le -{\log(1-x)}$ (valid for any $x\in(0,1)$) to $x=\frac1{n+k}$ and summing it from $k=0$ to $n$. – nejimban Oct 28 '23 at 11:56
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Does this answer your question? The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$ Found using Approach$0$, where you can find many other duplicates. – Anne Bauval Oct 28 '23 at 12:19

Limit of 1/n + ... + 1/2n

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