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

1 AnswersWhen 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

1 Asked on January 7, 2022 by lww

1 Asked on January 7, 2022 by ruby-cho

3 Asked on January 7, 2022 by lucas-g

numerical linear algebra ordinary differential equations proof explanation robotics

1 Asked on January 7, 2022 by kaba

1 Asked on January 5, 2022

3 Asked on January 5, 2022

algebra precalculus egyptian fractions elementary number theory factoring quadratics

4 Asked on January 5, 2022 by truth-seek

3 Asked on January 5, 2022

1 Asked on January 5, 2022

1 Asked on January 5, 2022

1 Asked on January 5, 2022 by confuse_d

2 Asked on January 5, 2022 by unreal-engine-5-coming-soon

1 Asked on January 5, 2022

0 Asked on January 5, 2022 by medo

fourier analysis fourier transform harmonic analysis real analysis

0 Asked on January 5, 2022 by bachamohamed

1 Asked on January 5, 2022 by jonathan-x

definite integrals improper integrals integration laplace transform

Get help from others!

Recent Questions

- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?
- How can I understand these variograms?

Recent Answers

- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- kjetil b halvorsen on How to test consistency of responses?
- Philipp on How do i draw a ray in unity
- eric_kernfeld on How to test consistency of responses?
- Justin Markwell on Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)

© 2022 AnswerBun.com. All rights reserved.