# Why does the integral test not apply to #Sigma (sinn/n)^2# from #[1,oo)#?

Because

If we choose naturally:

we can see that #f(x) is positive and infinitesimal, but not decreasing, so the conditions for the integral test are not met.

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The integral test applies to series of the form ( \sum_{n=1}^{\infty} a_n ) where ( a_n ) is a positive, continuous, and decreasing function. However, for the series ( \sum_{n=1}^{\infty} \left(\frac{\sin n}{n}\right)^2 ), the function ( \left(\frac{\sin n}{n}\right)^2 ) does not meet the conditions required for the integral test. Specifically, it fails to meet the condition of being continuous and decreasing over the entire interval ([1, \infty)). Since the function ( \frac{\sin n}{n} ) oscillates between positive and negative values for different integer values of ( n ), it does not satisfy the conditions for the integral test. Therefore, the integral test cannot be applied to this series.

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