If a full proof is really necessary, then it's alright to give one (I still don't that that should be necessary) but I just want to be able to justify that step so I can move on.
The question along with its solution are attached.
I'm stuck on the step where it compares the sum to the integral with the <= sign. I don't understand how someone can infer the truth of this in a way that requires no additional steps.
Basically, how can I show this to be true without requiring too much work? (I just want to grasp it intuitively - I don't need a full proof or anything).
Any help would be greatly appreciated!
Thanks in advance!
If a full proof is really necessary, then it's alright to give one (I still don't that that should be necessary) but I just want to be able to justify that step so I can move on.
I now see how an integral with the same limits (if that's the correct term) is less than (or equal to) the infinite sums based on the picture I'm attaching (which I got from Wikipedia) but what I'm almost understanding but not quite is how adding 1 to the beginning of the sum's limit makes it be less than (or equal to) the integral. I'm thinking it's because starting at k+1 rather than k makes the first rectangle in the Wikipedia example picture disappear? Assuming everything I said is right, what confuses me now is how can I know that that first rectangle's total area that was skipped via n=k+1 rather than n=k for the summation is larger than then every other rectangle's part above the curve?
Is it because I know 1/n^2 or 1/x^2 converges and therefore the little area over the curve per rectangle will converge to something smaller than the area of the first entire rectangle?