What is the procedure for testing convergence of alternating series?

In summary, the conversation discusses the concept of Alternating Series and its procedure for determining convergence. The conversation also includes two specific problems and their solutions using the alternating series theorem. Ultimately, the conversation concludes that both problems are convergent.
  • #1
noboost4you
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0
We just learned Alternating Series, but I am still unsure of the procedure. A couple of problems in the book have stumped me, and I need to test them for convergence.

1. Sum n=1 going to infinity of [cos(n*pi) / (n^(7/8))]
2. Sum n=1 going to infinity of (-n/5)^n

I am really lost, I'd really appreciate the help...
 
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  • #2
Okay, an alternating series converges if the absolute value of the terms in it approaches zero. It's basically as if you take a big step forwards, a smaller step back, an even smaller step forwards and so on. Try it yourself (but not in class or you'll get funny looks).

1) cos(n * pi) is 1 for n even and -1 for n odd. So your series is basically just

-1/(1^7/8) + 1/(2^7/8) -1/(3^7/8) + ...

All you have to do now is show that as n->infinity, 1/(n^7/8) -> 0, and you should have no problem with that.

2) Just rewrite this as ((-1)^n)/(5^n) and observe that (-1)^n is positive for even n and negative for odd n, so you have an alternating series. Then this question is pretty much the same as the last one.
 
  • #3
so for #1, i should check the lim as n -> infinity of bn and see if it equals 0? and bn in this case is [cos(n*pi) / (n^(7/8))] ... cause according to the alternating series theorem, if the limit = 0, then it is convergent...

havent gotten to #2 yet. stay tuned ;)
 
  • #4
Yeah, that's pretty much right. For these questions you need to a) show they're alternating and b) the limit as n -> inf is zero.
 
  • #5
ok, did them, let's see if they're right.

1) lim n->inf [cos(n*pi) / (n^(7/8))] = 1/inf == 0 therefore, the Sum is convergent

2) lim n->inf 5^(-n) == 0 and therefore, the Sum is also convergent

yes? no? thanks for your help
 
  • #6
Bingo.

Wasn't as hard as all that cos(n*pi) stuff made it look, eh? Just be glad they didn't ask you what the thing actully converged to; that's much harder.
 
  • #7
oh i know, thanks again for the help. really helped alot. I'm always pleased with the helpfulness this site brings.
 

1. What is convergence testing?

Convergence testing is a method used in scientific research to determine if a series of data or experimental results are approaching a certain value or outcome. It is used to assess the reliability and accuracy of the results.

2. Why is convergence testing important?

Convergence testing is important because it helps scientists determine the validity of their research and whether the results can be considered reliable. It also allows for the detection of errors or inconsistencies in the data, which can then be corrected to improve the accuracy of the results.

3. How is convergence testing performed?

Convergence testing is typically performed by analyzing a series of data points and determining if they are approaching a specific value or if they are consistently fluctuating around a certain point. This can be done through various statistical methods and calculations.

4. What are some common methods used for convergence testing?

Some common methods used for convergence testing include the ratio test, the root test, and the comparison test. These methods involve comparing the given series to a known convergent or divergent series to determine its behavior.

5. What are the limitations of convergence testing?

Convergence testing has some limitations, as it can only provide an indication of the convergence or divergence of a series, but it cannot guarantee it. Additionally, it may not be suitable for all types of data or experimental results, and alternative methods may need to be used in these cases.

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