They both get closer to but never reaching zero. Is the reason simply that one gets 1/x^2 gets closer to zero faster? So whats the threshold for something to be considered finite or infinite?
In: 59
The area under 1/x is closely related to the sum of all numbers 1/n which you can prove goes to infinity as follows.
You can sum the numbers in chunks:
* 1/2
* 1/3 + 1/4
* 1/5 + 1/6 + 1/7 + 1/8
* …
1/3 > 1/4, so 1/3 + 1/4 > 1/4 + 1/4 = 1/2.
1/5, 1/6, 1/7 are each greater than 1/8. So 1/5 + 1/6 + 1/7 + 1/8 > 4 * 1/8 = 1/2.
Each chunk has twice as many numbers as the previous, and each chunk sums to some value greater than or equal to 1/2.
For the sum to be finite, there is a number k such that that for all n, the sum is less than k.
So take a number k. If we sum up the first 2*k chunks (about 2^((2*k)) numbers) of the sequence, we get at least 2*k*(1/2) = k. So there is no limit to this sum.
If you try to do the equivalent for 1/(2^(x)) you get the following sequence:
1. 1/2
2. 1/2 + 1/4 = 3/4
3. 3/4 + 1/8 = 7/8
4. 7/8 + 1/16 = 15/16
And you can see that the nth number is always 1 – 1/(2^(n)), which will always be less than 1.
I used a different sequence there, because it’s a very neat one, but you get the idea I think.
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