# $\sum_{k=2}^{\infty} \left( \frac{1}{k-1} - \frac{1}{k} \right) = 1$

I'm reading a proof and very rusty. I'm missing something simple I suspect.

Have a little mercy and give me a nudge in the right direction?

$2\epsilon + \sum_{k=2}^{\infty} 2\epsilon \left( \frac{1}{k-1} - \frac{1}{k} \right) = 4\epsilon$

I can use $\sum k^{-2}=\frac{\pi^2}{6}$ to show that it is less than $6\epsilon$ which also works for the proof this is from, but how do I show it is exactly $4\epsilon$ since that is how the book does it? In other words how do I show:

$\sum_{k=2}^{\infty} \left( \frac{1}{k-1} - \frac{1}{k} \right) = 1$

I would prefer a hint.

• Take the partial sums and see what's getting canceled ... – rtybase Feb 4 '17 at 16:34
• So (if you're very rusty): $\frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4}... + \frac{1}{n-1} - \frac{1}{n}$ e.g. equals $1 - \frac{1}{n}.$ – Obriareos Feb 4 '17 at 16:38
• what a ridicolous downvote – tired Feb 4 '17 at 16:41
• @Obriareos Your comment is effectively the answer I posted. – Mark Viola Feb 4 '17 at 16:43
• @ tired: Downvote? @ Dr. MV: that's true - I haven't see your answer before. – Obriareos Feb 4 '17 at 19:11

Note that

\begin{align} \sum_{k=2}^K\left(\frac{1}{k-1}-\frac1k\right)&=\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right)+\cdots +\left(\frac{1}{K-1}-\frac{1}{K}\right)\\\\ &=1-\frac1K \tag 1 \end{align}

More formally, note that if $S_K=\sum_{k=2}^K\left(\frac{1}{k-1}-\frac1k\right)$, then $S_{K+1}=S_K+\frac{1}{K}-\frac{1}{K+1}$. So, we can use induction to prove the result given by $(1)$.

• Hehe ...too many very similar to a basic question. I'm deleting mine...+1 – DonAntonio Feb 4 '17 at 16:36
• @DonAntonio Yes indeed. – Mark Viola Feb 4 '17 at 16:37

Hint:

hope you can take it from here.

As @CarstenS pointed out the picture shown here is somehow a more correct hint

• Now that deserves a (+1) just for the creativity! – Mark Viola Feb 4 '17 at 16:38
• @Oliver821 very good! – tired Feb 4 '17 at 16:38
• @Dr.MV thank you very much! :) – tired Feb 4 '17 at 16:39
• Yes the hint should be obvious by now. Hint: Split the sums and solve each telescoping series individually. – John Bradshaw Feb 4 '17 at 16:39
• And this put a big smile on my face. Nice picture by the way. – Mark Viola Feb 4 '17 at 16:43

It's a telescoping series.

First show:

$$\sum_{k=2}^N \frac{1}{k-1}-\frac{1}{k}=1-\frac{1}{N}$$

Fix $n\geq 2$. Then $$\sum_{k=2}^n\left(\frac{1}{k-1}-\frac{1}{k}\right)=\sum_{k=2}^n \frac{1}{k-1}-\sum_{k=2}^n\frac{1}{k}=\sum_{k=1}^{n-1}\frac{1}{k}-\sum_{k=1}^n\frac{1}{k}=1+\sum_{k=2}^{n-1}\frac{1}{k}-\sum_{k=2}^{n-1}\frac{1}{k}-\frac{1}{n}=1-\frac{1}{n}.$$ Let $n\to \infty$ and you'll get your result.

Well it has been done before, but I still like to give another method:

$$\sum_{k=2}^{m}\frac{1}{k-1}-\frac{1}{k}=\sum_{k=1}^{m-1}\frac{1}{k}-\sum_{k=1}^{m}\frac{1}{k}+1$$$$=\left[\sum_{k=1}^{m-1}\frac{1}{k}-\ln\left(m-1\right)\right]-\left[\sum_{k=1}^{m}\frac{1}{k}-\ln\left(m\right)\right]+1+\ln\left(\frac{m-1}{m}\right)$$

Taking the limit of the expression yields:

$$\underbrace{\left[\lim_{m\to\infty}\sum_{k=1}^{m-1}\frac{1}{k}-\ln\left(m-1\right)\right]}_{\gamma }-\underbrace{\left[\lim_{m\to\infty}\sum_{k=1}^{m}\frac{1}{k}-\ln\left(m\right)\right]}_{\gamma }+1+0=\color{red}{1}$$

Where $$\gamma$$ is Euler–Mascheroni constant.