Calc6.5

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Explain why the integral test is or is not applicable to \sum _{{n=1}}^{{\infty }}n^{2}.

The function f(x)=x^{2} is positive and continuous on the interval [1,\infty ]. However, it is increasing whereas the integral test only works on functions which are decreasing. The divergence of this series, though, is easily established. Any continuous function which is positive and increasing can not go to 0. If the terms of a series do not go to 0, then the series diverges by the nth term test.


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