calculus - Is there any method to get a finite sum for $1+frac{1}{2}+frac{1}{3}+frac{1}{4}+cdots$?

As we can see on Wikipedia, there are some algebraic methods that give us finite sums for the Grandi's series

$$1-1+1-1+1-1+1-1+\cdots$$

Let $S$ be the sum of the Grandi's series. Then

• $S=(1-1)+(1-1)+\cdots=0+0+\cdots=0$




• $S=1+(-1+1)+(-1+1)+\cdots=1+0+0+\cdots=1$




• $1-S=S$ so that $S=1/2$

The algebraic manipulations above are not allowed because the Grandi's series doesn't converges.

Is there something similar related to the harmonic series? In other words, is there any "incorrect algebraic way" to get a finite sum for

$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots \;?$$

Thanks.

Answer

Actually, if you only consider the normal harmonic series, with only positive terms, you can't find a rearrangement that will converge to a certain value since all terms are positive, and changing the order is not possible.

However you can prove that, if you re-arrange correctly the terms of the series of general term: $a_n = \frac{(-1)^{n-1}}{n}$ , you can make it converge towards any real x.

Roughly: $a_1 = 1$ , if x<1, add negative terms until $x> S_{f(n)} $ where this partial sum is made with odd terms of the series. Then once you have this condition, you will add positive terms until : x < $S_{f(n)} $ . You do that undefinitely, and putting it properly enough you can prove that this series converges towards x. Of course this in only sketches, but the idea is here