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Numerade Educator

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Problem 28 Easy Difficulty

Explain why the Integral Test can't be used to determine whether the series is convergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\cos^2 n}{1 + n^2} $

Answer

It is because $f(x)=\frac{\cos ^{2} x}{1+x^{2}}$ is not monotonically decreasing.

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Video Transcript

they're three main conditions for integral test. We need the function to be continuous, any function to be positive. I mean it function the decreasing now dysfunction is definitely continuous, and because the coastline is being squared is also always going to be positive. However, we need to know if it's decreasing. And unfortunately for us, if we have our function to find, we run into a problem because I take, let's say of hi over too. L'LL give me zero. However, if I plug in if a pie we'LL get one over one plus hi squared, which is greater than zero a pie. It's greater than pie over too. So that means the function. He is not always decreasing, so closure is not decreasing. It fails that requirement.