[Help!!!][Calculus - Integrals] How do I prove this?

Question

Show that (sigma from k=1 to infinite)1/k is infinite but that (sigma from k=1 to infinite)1/k^2 is finite.

*instead sigma notation which I was not able to place here, I indicated it with (sigma...)

I got this answer from someone here. but I was told this is not enough proof. I think the transition from sigma to integration is not fully explained. can someone help?

take the anti-derivative of the first summation and you get:

ln|k| from 1 to infinite

this is equal to infinite.

take the anti-derivative of the second summation and you get:

1 / -k from 1 to infinite

this would be equal to:

(-1/infinite) - (-1/1) = 1

Answer 1

These are quite famous problems. As you mentioned, you can do it with integration; thats probably the simplest way.

In order to see why that works, lets call f(x) = 1/x or 1/x^2 depending on which one you want.
Draw a 'histogram', where each of the bars is 1 wide, and has height 1/(left x value). So you have a bar from x=1 to x=2 with height 1, x=2 to x=3 with height 1/2, etc.

Now if you draw the curve f(x), it never goes below any of these bars. So the integral of f(x), which is the area under the curve, must be less than the total area of each bar, which is that infinite sum. So if we get a finite integral, we have must a finite sum.

Similarly, you could draw a histogram where each bar has height 1/(right x value). Then the curve is always *below* the bars, so you get a lower limit. Then if that integral is infinite, the area must be infinite.

Anyway, now that we've got the official answer sorted, there are many other proofs. Heres a nice one for the 1/k case:
= 1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ...
>= 1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ...
= 1 + 1/2 + 1/2 + 1/2 + ...
which goes to infinity.

As a side note, the actual sum of 1/k^2 is pi^2/6. That always interests me - if you draw a whole bunch of circles with radius 1, 1/2, 1/3, .., then their total perimeter turns into the 1/k-type sum, and is infinite. However, their total area turns into the 1/k^2-type sum, and is finite. So you can't draw the outline of the circles, but you can colour them in.. ;)

Answer 2

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