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

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Answer 1 · 2016-12-28T18:04:43

Because the condition only applies if #f(x)# and #a_n=f(n)# are non-negative and monotone decreasing, which is not the case here.

In fact, if we choose:

#f(x) = (2+sinx)/x#

we have that the function is continuous and non-negative for #x>0#, but as:

#f'(x) = frac(xcosx-2-sinx) (x^2)#

is positive for some values of #x#, clearly #f(x)# is not monotone decreasing.

graph{(2+sinx)/x [-10, 10, -5, 5]}