This section relies on a fair bit of calculus. We'll start by looking at the p-integral.
The integral is a simple one (I've flipped the exponent and denominator of the integral from -p+1 to 1-p for convenience).
Now of course this is an improper integral, so we have to evaluate it using the limit as our "dummy" variable R approaches ∞ .
The second term above is finite because a is a constant. The first is trickier. Depending on the value of p, it will either converge to a limit or diverge to infinity.
Here's the full limit for the case where p > 1:
and for p < 1:
We already said that when p = 1 the integral is ln|x| + C, which approaches ∞ as R→∞. With the p-integral in hand, we can use the integral test to determine which p-series converge.
To determine whether the harmonic series converges, we let f(x) = 1/x and integrate the function from 1 to ∞. Clearly the series is trapped under f(x) as drawn, so if the function converges to a finite limit, so must the series. On the other hand, if the function diverges, then the series won't converge to a fixed limit either.
This integral is the special case of the p-integral with p = 1. It goes like this:
Because the integral that traps the series below it does not reach a finite limit (diverges), the series diverges.
It might be a little difficult to come to terms with the notion that the harmonic series doesn't eventually sum to a limit even thought the limit of the nth term is zero, but it's true.
The denominator of the terms of the harmonic series just doesn't get large "fast enough" for the series to converge.
On the right is a comparison of the sums
The left column of each colored block is just the nth term. The right column is the cumulative sum. When a series converges, it is clear that successive terms modify digit positions more and more to the right of the decimal. That's not true of the harmonic series.
The graph on the right shows how each of these sums grow. The convergent p-series approaches a horizontal asymptote near y = 1.6. While the growth of the harmonic series slows down as x grows, the series is unbounded, and will continue to grow without converging to a finite limit.
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