Hello! Yesterday we saw an infinite process that
approached something exact. 1 half plus 1 fourth
plus 1 eighth plus 1 sixteenth plus dot dot dot.
Even though it continued forever, the sum approached
1 perfectly. Today, let's consider something
that seems similar, but behaves completely different.
Here is a sum that we refer to as the harmonic
series. 1 over 1 plus 1 over 2. plus 1 over 3,
plus 1 over 4, plus 1 over 5, plus dot dot dot.
Just like yesterday, the terms keep getting smaller.
And because of that, it is tempting to think
this sum should also settle down to some finite
number. But it doesn't. Even though the terms
shrink, they do not shrink fast enough. And as
we keep adding more and more terms, The total
continues to grow. Slowly. Very slowly. But it
never settles. It keeps increasing without bound.
To infinity. The infinite sum from yesterday
and today look almost the same. In both examples,
we keep adding forever, the terms get smaller,
and the process never ends. What I find interesting
is one infinite sum approaches a finite number,
1, and the other grows not only to 2 or 3 or
even 100, but without bound. In mathematics,
we say the first series converges and the second
diverges. A small change in the pattern creates
a completely different outcome. But the part
I find most interesting is that this sum grows
unbelievably slowly. To reach a total of 2 only
requires 4 terms. To reach 3 requires 11 terms.
To reach 4 takes 31 terms. Already, the slowdown
is evident. But then it becomes dramatic. To
reach 10, requires more than 12 ,000 terms. And
to reach 100? Well, that requires an unimaginably
enormous number of terms. At every stage, it
feels like the sum is barely moving. You might
think, surely, eventually, it must level off.
But it never does. No matter how large a number
you choose, this sum will eventually grow larger
than that number. Now, my intuition is this seems
impossible. But infinity has a different ending,
which makes infinity feel even stranger. This
is not merely something mathematicians suspect
to be true. In mathematics, we can actually prove
it with certainty. No matter what positive number
we imagine, this infinite sum will eventually
surpass it, even though the number of terms required
may itself feel beyond imagination. Well, if
you want to understand why the harmonic series
continues without bound, you can read my article
called A Log That Is Natural. It's on my Lazarus
Math series at intersectingus .com. Okay, tomorrow
we'll take a different perspective on infinity
to see where it appears in surprising places.