Alternating Series Convergence: Is the Series Sum of sin(1/n^2) Convergent?

In summary, the conversation discusses the convergence of the series sum of sin(1/n^2) where n goes from 1 to infinity. The suggested approach is to use the limit comparison test with a similar series b_n = 1/n^2. The limit of a_n/b_n is shown to be finite, implying the convergence of the series sum of a_n. The conversation also addresses a potential error in considering sin as the denominator instead of the numerator.
  • #1
CalculusHelp1
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Homework Statement



Sorry I don't know how to use symbols on this site so bear with me:

the question is does the following series converge: sum of sin(1/n^2) where n goes from 1 to infinity

Homework Equations



Limit comparison test, maybe others

The Attempt at a Solution



Okay I think I got the solution but I'm not sure if the logic is correct. I divided sin(1/n^2) by the p-series (1/n).

This comes out to n/sin(1/n^2)...the top will go to infinity and the bottom will go to 0 as n goes to infinity, so the series divergies. Because 1/n also diverges (harmonic series) sin(1/n^2) also diverges. Is this right?
 
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  • #2
Don't know why I said 'alternating' series in the thread title...my brain is becoming mush!
 
  • #3
I think limit test is probably the way to go, however I think you're doing a few things wrong with it.

With [tex]a_n[/tex] being the given function, consider [tex]b_n = \frac{1}{n^2}[/tex]

Then,
[tex]\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} n^2\sin{\frac{1}{n^2}} [/tex]

Now, you should be able to show that that limit is finite, so the convergence of the sum of [tex]b_n[/tex] would imply the convergence of the sum of [tex]a_n[/tex]. Does that help?
 
  • #4
Yep that fixes it, I also made an algebraic error thinking sin was on the denominator. Brain is going to mush. Thanks!
 

What is an alternating series?

An alternating series is a type of mathematical series in which the terms alternate between positive and negative numbers.

What is the alternating series test?

The alternating series test is a method used to determine the convergence or divergence of an alternating series. It states that if the terms of an alternating series decrease in absolute value and approach zero, then the series converges.

How do you determine the convergence of an alternating series?

To determine the convergence of an alternating series, you can use the alternating series test or other convergence tests such as the ratio test or the root test. These tests can help determine if the series converges, diverges, or is inconclusive.

What is the difference between absolute convergence and conditional convergence?

Absolute convergence occurs when a series converges regardless of the order in which the terms are added. In contrast, conditional convergence occurs when the series only converges if the terms are added in a specific order.

Can an alternating series converge to a non-zero value?

Yes, an alternating series can converge to a non-zero value. For example, the alternating series (-1)^n/n converges to ln(2) even though the terms approach zero. However, the alternating series test only applies to series that converge to zero.

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