## Harmonic Series

The harmonic series is this:

1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+...

Some infinite series sum to real numbers (see Geometric Series). These 9 terms sum to 2.829. 100 terms sum to 5.187. 1000 terms sum to 7.4854. And 1,000,000 terms sum to 14.384. Just what does the infinite series add up to? The answer to that is that the sum blows up to infinity. It gets there very slowly, doesn't it? There is actually a simple proof that it sums to infinity:

S=1+(1/2)+(1/3+1/4)+(1/5+1/6+1/7+1/8)+(1/9+...
Consider this series:
T=1+(1/2)+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16+...
T<S for any finite number of terms.
T=1+1/2+1/2+1/2+... which is infinite for infinitely many terms.
So S must also be infinite.

Just what is the sum of the first n terms?

S(n)=1+1/2+1/3+1/4+1/5+...+1/n
S(1)=1
S(2)=(2+1)/2!
S(3)=(3!+3+2!)/3!
S(4)=(4!+4x3+4x2+3!)/4!
S(5)=(5!+5x4x3+5x4x2+5x3x2+4!)/4!
The stuff in parentheses forms the following pattern:
S(5)=(5!+5!/2+5!/3+5!/4+5!/5)/5!
We can divide top and bottom by 5! and get:
S(5)=1+1/2+1/3+1/4+1/5
Oops!

That's what we started with. All I did was show that S(5)=S(5). Actually, we can go directly from the last step (just above the Oops!) to the step above that with all of the 5! in it. Just multiply top and bottom by 5!

It turns out that there is a relatively simple estimate of the sum of n terms:

S(n) is approximately ln(n)+.5772156649...

It gets closer, the larger n is. The number .5772156649... is known as Euler's constant (or the Euler-Mascheroni constant), and is represented by a small gamma. And ln is the natural logarithm (the base e logarithm). See Logarithms. e is Euler's number and is 2.7182818284590... I will write articles about e and logarithms eventually.

There are other series which sum to infinity, which have even smaller terms than this harmonic series. The really famous one is this:

1/2+1/3+1/5+1/7+1/11+1/13+...

Here all of the denominators are primes (See Sieve Of Eratosthenes). And this too sums to infinity. Once this is proved, it follows that there are infinitely many primes, which we already knew (See The Infinitude Of Primes). That doesn't work backward; infinitely many primes does not imply that this series sums to infinity. This series builds up to large numbers much more slowly than does the harmonic series. After a million terms, the sum is only 3.068.