Does the series \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) converge?

HeDi's summary: In summary, the problem asks to determine whether the series \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) diverges, converges conditionally, or converges absolutely. The nth term test shows that the limit of the sequence is 0, indicating that the function does not necessarily diverge. The integral test and root test cannot be applied, and the ratio test does not work. The approach suggested by BiPHedi is to use limit comparison with 1/n^4.
  • #1
Bipolarity
776
2

Homework Statement



Determine whether the following series diverges, converges conditionally, or converges absolutely.

[tex] \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) [/tex]

Homework Equations


The Attempt at a Solution


This was on today's test, and was the only problem I wasn't able to solve. I doubt my teacher will be going over these, and in any case his explanations never satisfy me, so could someone help me with this?

According to the nth term test, the limit of the sequence is 0, since sin(x) is continuous, so the function doesn't necessarily diverge.

Integral test can't be applied because the sequence is not monotonic. Root test serves no purpose. Limit comparison might work, but with what? Ratio does not work (I think?).

How might I approach this?

BiP
 
Physics news on Phys.org
  • #2
Hedi beat me to it down below VVV haha. I forgot that something bigger that diverges tells you nothing.
 
Last edited:
  • #3
use limit comparison with 1/n^4.for large n this is a positive series.
 
  • #4
hedipaldi said:
use limit comparison with 1/n^4.for large n this is a positive series.

Thanks!

But damn! I wish I thought of that in the test! Oh well.

BiP
 

FAQ: Does the series \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) converge?

What is series convergence?

Series convergence is a mathematical concept that refers to the behavior of an infinite series, which is a sum of an infinite number of terms. It determines whether the series will reach a finite value (converge) or approach infinity (diverge).

How do you test for series convergence?

There are several tests that can be used to determine series convergence, including the ratio test, the root test, the integral test, and the comparison test. These tests involve manipulating the terms of the series and analyzing their behavior to determine if the series converges or diverges.

What is the difference between absolute convergence and conditional convergence?

Absolute convergence occurs when the absolute value of the terms in a series converges, while conditional convergence occurs when only the terms themselves converge. In other words, a series with absolute convergence will always converge, but a series with conditional convergence may converge or diverge depending on the specific values of the terms.

Can a series converge to a value other than zero?

Yes, a series can converge to a value other than zero. This is known as a non-zero limit, and it means that the series approaches a specific finite value as the number of terms in the series increases. A series can also diverge to infinity, where the terms of the series become increasingly larger without approaching a specific value.

Why is series convergence important?

Series convergence is important in many areas of mathematics and science, as it allows us to determine the behavior of infinite series and make predictions about their values. It is also used in various applications, such as in engineering, physics, and finance, to analyze and model real-world phenomena.

Similar threads

Proving convergence and divergence of series
Replies
3
Views
798
Interval of convergence of power series from ratio test
Replies
2
Views
787
How to show that these sequences are summable?
Replies
1
Views
988
Convergence of oscillatory/geometric series
Replies
1
Views
1K
Prove that this series converges uniformly on R
Replies
4
Views
982
On pointwise convergence of Fourier series
Replies
3
Views
1K
Doubt regarding the series ##\sum [\sqrt{n+1} - \sqrt{n}\,]##
Replies
17
Views
2K
Determining if a series converges
Replies
4
Views
1K
On the ratio test for power series
Replies
2
Views
1K
Root test proof using Law of Algebra
Replies
6
Views
1K

Hot Threads

Recent Insights

Back
Top