Proving Convergence of an Infinite Series

In summary: It's been a few decades, but I suppose that the integral test is still taught in freshman calculus. The advantage of using the integral test is that in addition to convergence, since the bounding integral goes to zero as x to infinity, you also get that the series goes to zero.
  • #1
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Okay, I have this function defined as an infinite series:

[tex]f(x) = \sum_{n=1}^{\infty}\frac{\sin(nx)}{x+n^4}[/tex]​

which is converges uniformly and absolutely for x > 0. I have shown that f is continuous and has a derivative for x > 0. Now I have to show that [tex]f(x) \rightarrow 0[/tex] as [tex]x \rightarrow \infty.[/tex] It's obvious that it is the case, but how do I prove it. I've tried putting 1/x outside the sum, but then I don't know about the remaining part. Any ideas?
 
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  • #2
set [tex]f_n(x) = \frac{sin(nx)}{x+n^4}[/tex], then [tex]|f_n(x)| \leq \frac{1}{n^4}[/tex] for all x>0
 
  • #3
Hmm.. How does that show that f(x) -> 0 as x -> infinity?
 
  • #4
broegger said:
Hmm.. How does that show that f(x) -> 0 as x -> infinity?
by the comparison test
 
  • #5
lurflurf said:
by the comparison test

Maybe I'm a little slow here, but [tex]\sum_{n=1}^{\infty}\frac1{n^4}[/tex] does not equal 0, so how come?
 
  • #6
broegger said:
Maybe I'm a little slow here, but [tex]\sum_{n=1}^{\infty}\frac1{n^4}[/tex] does not equal 0, so how come?
My mistake. Have you any theorems concerning the interchange of limits for uniformly convergent series?
 
  • #7
No, I can't find any relevant theorems to apply. I'm lost at sea.
 
  • #8
fourier jr said:
set [tex]f_n(x) = \frac{sin(nx)}{x+n^4}[/tex], then [tex]|f_n(x)| \leq \frac{1}{n^4}[/tex] for all x>0

This series converges absolutely by the p-series test. I don't see why you have to look for any limits or to check it is decreases. If it comverges absolutely, then it converges.

Regards,

Nenad
 
  • #9
[tex]f(x) = \frac{sin(x)}{x+1} + \sum_{n=2}^{\infty}\frac{\sin(nx)}{x+n^4}[/tex]

This is clearly bounded by:

[tex]\frac{1}{x+1} + \int_{n=1}^\infty \frac{1}{x+n^4}\,\,dn[/tex]

which can be shown to go to zero.

Carl
 
  • #10
isn't that a much harder way to do it though? & judging by brogger's other posts i would say he's in the standard advanced calculus course where you'd learn the usual tests for convergence. (like comparison)
 
  • #11
It's been a few decades, but I suppose that the integral test is still taught in freshman calculus. The advantage of using the integral test is that in addition to convergence, since the bounding integral goes to zero as x to infinity, you also get that the series goes to zero.

Carl
 

What is an infinite series?

An infinite series is a sum of infinitely many terms, where each term has a specific pattern or relationship with the previous terms. It is written in the form of a1 + a2 + a3 + ... + an + ...

How is the convergence of an infinite series determined?

The convergence of an infinite series is determined by examining the behavior of the partial sums, which are the sums of the first n terms of the series. If the partial sums approach a finite value as n increases, then the series is said to converge.

What is the difference between absolute convergence and conditional convergence?

Absolute convergence refers to a series where the individual terms are positive, and the series converges regardless of the order in which the terms are added. Conditional convergence refers to a series where the terms alternate in sign, and the series only converges if the terms are added in a specific order.

What are some common tests used to prove the convergence of an infinite series?

Some common tests for convergence include the comparison test, limit comparison test, ratio test, and root test. These tests compare the given series to a known convergent or divergent series to determine the behavior of the given series.

Are there any special cases where a series may not converge or diverge using these tests?

Yes, there are some special cases where the tests may not be applicable or conclusive. These include series with terms that do not approach zero, series with alternating signs that do not satisfy the alternating series test, and series with terms that approach zero too slowly for the tests to determine convergence or divergence. In these cases, other methods, such as the integral test or the alternating series test, may be used.

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