Does the harmonic series diverge or converge?
Solution
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.1/20 + 1/21 + 1/22 + 1/23 + 1/24 + … = 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 2
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + … = ∞
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. 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 + …
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 + …
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) + …
1/1 + 1/2 + 1/2 + 1/2 + 1/2 + …
1/1 + 1/2 + 1/2 + 1/2 + 1/2 + … = ∞
1/1 + 1/2 + 1/2 + 1/2 + 1/2 + … = ∞
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 + …
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:
However, the harmonic series doesn’t:
This is very hard to get for most people. I mean, look at this:
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.
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:
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) + …
Which simplifies to:
Which clearly diverges to infinity:
So if
and
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
Source: thisinsider.com