Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I've been reviewing some calculus material lately and I just have a couple questions:

1) I've seen infinite series shown graphically as a collection of rectangular elements under a curve representing an approximation of the area under the curve. But the outputs of the infinite series are just one dimensional numbers, so wouldn't lines coming from each natural number extending upward to the functional value of the curve be a more realistic graph? How do we get rectangles when we are summing numbers? We don't have a value for n=1.5 when the sum is from n=1 to n= infinity, so why is there rectangular area at x = 1.5 on the graph? (I know I'm wrong-- I just want to know why I am wrong.)

2) I came to my first question by noticing that the infinite series 1/x^{2}converges to pi^{2}/6 while its associated improper integral converges to 1. I guess I feel like whatever differences exist in these values would be "straightened out" or made trivial because we are sending x to infinity.

Thanks in advance!

-Lee

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# Infinite series and improper integrals

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