Study the convergence of the series $sum_{n ge 1} sin frac{1}{n^{5/4}}$.

I have to show whether the series

$$sum_{n ge 1}sin frac{1}{n^{5/4}}$$

is convergent or not.

This is what I tried, but I am not sure if it’s correct:

We know:

$$sin x le x, hspace{1cm} forall x ge 0$$

So then

$$hspace{6cm} sin frac{1}{n^{5/4}} le frac 1{n^{5/4}}, hspace{.5cm} forall n in mathbb{N} hspace{3cm}(1)$$

By the generalized harmonic series, we also know

$$hspace{7cm} sum_{n ge 1} frac{1}{n^{5/4}} hspace{.25cm} text{convergent} hspace{3cm} (2)$$

Now, using $$(1)$$ and $$(2)$$, we can conclude by the First Comparison Test that the series

$$sum_{n ge 1} sin frac{1}{n^{5/4}}$$

is convergent.

Is this correct?

Mathematics Asked by user592938 on December 31, 2020

When you wrote that $$displaystylesinleft(frac1{n^{5/4}}right)leqslant n^{5/4}$$, what you should have written was that $$displaystylesinleft(frac1{n^{5/4}}right)leqslantfrac1{n^{5/4}}$$.

Besides, the comparison test is for series of non-negative numbers. So, you should add to your proof that$$(forall ninBbb N):frac1{n^{5/4}}inleft(0,fracpi2right)implies(forall ninBbb N):sinleft(frac1{n^{5/4}}right)>0.$$

Correct answer by José Carlos Santos on December 31, 2020

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