Formula for $\sum_{k=1}^n \frac{10}{10+k}$

Question

In a normal problem solving I run into a sum:

$\sum_{k=1}^n \frac{10}{10+k}$

Let $n=20$ or something like that (not huge). Browsing a list of sums and another, I'm not finding a formula for this one.

I'm wondering is it possible to find a formula for this type of sum, or if this type of sum has to be computed by a computer summing it.

http://math.stackexchange.com/questions/5035/1-1-2-1-3-1-n – Aryabhata Jan 15 '11 at 22:39 — Aryabhata, Jan 15 '11 at 22:39

Ross Millikan · Accepted Answer · 2018-09-25T13:44:31.077

8

You can express it using the Harmonic numbers as $\sum_{k=1}^n \frac{10}{10+k}=10(H_{10+n}-H_{10})\approx 10(\ln(10+n)-\ln(10))$ But if you want it exactly you need to add it up.

edited Sep 25 '18 at 13:44

answered Jan 15 '11 at 15:32

Ross Millikan

374,822

4

I evaluated $$\sum_{k=1}^{n}\frac{10}{10+k}=10\sum_{k=1}^{n}\frac{1}{10+k}=10\sum_{k=11}^{10+n}\frac{1}{k}=10\left( H_{10+n}-H_{10}\right) $$ – Américo Tavares Jan 15 '11 at 15:37
@Americo: You are right. I didn't read carefully. Fixed – Ross Millikan Jan 15 '11 at 15:41

Formula for $\sum_{k=1}^n \frac{10}{10+k}$

1 Answers1