Tommi pointed out in the comments to the previous post that I had made a mistake in my purported proof that
because while finding the limit, I assumed the limit exists. It made me sad, because I thought the proof was really cute. But actually it was just incomplete. It feels just like that time when I was a kid and we brought home new puppy, only to discover it was missing a leg. My sister and I made a new leg out of Tinker Toys and sewed it on there, despite the puppy’s vociferous protests. I figure the same basic strategy is in order now, so let’s re-examine that expression.
We can verify the limit in a straightforward way using Hopital’s Rule
For more on Hopital see the book by Gilbert Strang, (section 3.8).
One way to build some intuition would be to make some plots. Here’s (red) and (blue) plotted together.
So if is getting bigger and bigger faster than , the numerator of the fraction is growing faster than the denominator, and the limit should increase up to infinity. But it looks instead like actually grows faster. Then the denominator is growing faster than the numerator, and we expect the limit to be zero.
Here’s a plot of the actual thing, .
Definitely going towards zero.
Finally, one more proof. I want to take a look at the series expansions and see if the limit of that is obvious as goes to zero. The best place to take a series of logarithm is around .
We want the limit as , which is the same as the limit as . Now there are no zeroes or infinities any more, so we can just plug straight into the last line of the equalities above.
Let’s do the summation. The first few terms to add are
and the first few partial sums are
From this we conjecture that
It’s been shown for already. Use induction, assuming it’s true for , and showing it’s true for .
Which proves the conjectured summation. The limit of the sum as is
With this result for the summation, we just get