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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.

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up vote 7 down vote accepted

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.

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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
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