Posted by Alex on .
Explain clearly in detail the difference of the summation of n=1 to infinity of 1/n^2; and the integral of 1/x^2dx with the limit of 1 to infinity, as well.
Part 2: Explain the natures of the two expressions and explain why the summation is greater than the integral test.
Any help would be greatly appreciated
Calc 2, Series/Sequences -
Drawa graph of the function
f(x) = 1/x^2.
f(1) + f(2) + f(3) + ...
can be intepreted as the total area of the rectangles of heights f(1), f(2),
f(3),... and all of width 1.
If you draw these rectangles in your graph, the one with height f(1) is between x = 1 and x = 2, the rectangle of height f(2) is between x = 2 and x = 3, etc., then you clearly see that the total area of all the rectangles together is less than the area under the curve, due to the fact that f(x) is a decreasing function.
THe precise relation between the summation and the integration is given by the Euler–Maclaurin summation formula: