# Explicit Formula for Sum of Series

## Homework Statement

Determine an explicit formula for the sum of n terms for the given series:

1, $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$, $$\frac{1}{5}$$

## The Attempt at a Solution

I calculated the first 5 terms for the sum sequence and got:

1, $$\frac{3}{2}$$, $$\frac{11}{6}$$, $$\frac{25}{12}$$, $$\frac{137}{60}$$

but I can't find a pattern to determine an explicit formula.

Maybe I'm misunderstanding your question, but wouldn't it just be $$\sum_{n=1}^\infty 1/n$$ ?

The question asks for n terms, not an infinite number of terms

Oh, sorry. I read the original post too quickly.

That's alright, nothing to bash heads in about! As for the problem, it's a doozy...

You're right, it is a doozy! This series is known as a harmonic series, and according to http://plus.maths.org/issue12/features/harmonic/index.html" [Broken], "there is no simple formula, akin to the formulae for the sums of arithmetic and geometric series, for the sum."

Last edited by a moderator:
Well then wouldn't it just be:

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

$$S_1=1$$

$$S_2=1+\frac{1}{2}=\frac{2+1}{2}$$

$$S_3=1+\frac{1}{2}+\frac{1}{3}=\frac{6+3+2}{6}$$

$$S_4=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}=\frac{12+6+4+3}{12}$$

blah, that doesn't actually seem to work!!

I know! It's whack!

I started out trying to find formulas for numerator and denominator separately. In cases where it doesn't simplify, the denominator is n! . As for the numerator, aside from the obvious (n! + n!/2 + n!/3+...+n!/n) I do not see how to arrive at a formula.

Dick
Homework Helper
According to http://mathworld.wolfram.com/HarmonicSeries.html you can write it as a sum of the Euler-Mascheroni constant and a digamma function. I'm guessing if there were an easier expression they would have mentioned it. I think steelphantom is right, there is no elementary formula.

I think all it is is sigma from k=2 to n of 1/n. very simple that would start off with one half then one third then one quarter and so on.

Tom Mattson
Staff Emeritus
Gold Member
That much is obvious. The trouble is finding a formula for the nth partial sum.

I think it's impossible, infinity would be the common denominator.

Mark44
Mentor
I think it's impossible, infinity would be the common denominator.
Not for the sum of n (a finite number) terms.

I think it's impossible, infinity would be the common denominator.

The denominator ought to be the lcm(1,2,...,n) whereas the numerator is the puzzle.