# Summing The Following: $\ 1, 1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$

I can't seem to get this to sum, I'd be very thankful if someone could help me out.

$$\ 1, 1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$$

NOTE It would be nice if someone would post a consistent 'algorithm' / method that always works for sequences of this form that can be summed. As I find them very interesting and they've come up a lot for me lately....

• The correct search terms are "Harmonic sum" and "Harmonic Series" – JMoravitz Sep 29 at 16:05
• NOTE Sorry, to everyone, I just realized I put the wrong code in the header..... Thanks to @N. F. Taussig for fixing it! – Jinny Ecckle Sep 29 at 16:09
• @JMoravitz, I think an issue I'm having, now that I looked at it, is the form I've written it in. – Jinny Ecckle Sep 29 at 16:20
• I don't get if you want a closed form for $H_n$ or for $\sum_{k=1}^{n}H_k$. In the latter case, you may just invoke summation by parts. – Jack D'Aurizio Sep 29 at 16:30
• For ∑nk=1Hk. Thanks for taking the time! – Jinny Ecckle Sep 29 at 16:32