Harmonic series

This is the harmonic series:
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + …
The denominators continue to increase to infinity.

Many other infinite series converge to a single number:

1/2⁰ + 1/2¹ + 1/2² + 1/2³ + 1/2⁴ + … = 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 2

However, the harmonic series doesn’t:

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + … = ∞

This is very hard to get for most people. I mean, look at this:

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + …
It seems like it’s slowing down! Look at how little the fractions get. And they’re only getting smaller! But the thing is, the harmonic series doesn’t converge to a single number. It goes to infinity. Very, very slowly.

Let’s prove it. Let’s compare the harmonic series to another, smaller series.

       Harmonic series:         1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + …
Obviously smaller series: 1/1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + …

Notice how, following the second term, every number in the second ifinite series is smaller than the same term in the Harmonic series. So:

      Harmonic series                     Obviously smaller series
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + …    >    1/1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + …

But let’s look at that second infinite series:

1/1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + …
1/1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + …

Which simplifies to:

1/1 + 1/2 + 1/2 + 1/2 + 1/2 + …

Which clearly diverges to infinity:

1/1 + 1/2 + 1/2 + 1/2 + 1/2 + … = ∞

So if

1/1 + 1/2 + 1/2 + 1/2 + 1/2 + … = ∞

and

      Harmonic series                     Obviously smaller series
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + …    >    1/1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + …

then

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + … = ∞

Source: thisinsider.com