The Integral Test

Alex Egg,

What is an integral - it's the area under a curve (simply). Or the reiman sum from x to xn of a function. Now think about what a series is - it's the sum of all the partial sums of a sequence. What is a sequence - a series a list of numbers I guess - or you can think of it as a continuous function. If you think of a sequence as a function and a series as the sum of each sequence element than isn't that just saying the x and y value for each part of the sequence. So were adding up each function value for the sequence - doesn't that sound just like the reiman sum of a continious function. Therefore for a continuous  function (w/ non-neg values) the sum of the series/convergence point (if it converges - see convergence tests) is the integral of the series. 

 

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Tags: math, calc, integral

Last edited by Alex Egg, 2016-10-05 19:25:12
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