Convergent or Divergent? Alternating Series Help for Tomorrow's Test

In summary, the conversation is about the convergence of an alternating series and the methods used to determine its convergence. The root test and ratio test were not applicable and even Maple 10 could not provide an answer. The possibility of using the alternating series test is mentioned, but it is pointed out that the terms in the series do not approach zero and therefore the series cannot converge. The speaker also suggests that the series may be bounded but still does not converge to a well-defined limit point. The importance of the terms approaching zero for a series to converge is mentioned.
  • #1
SigurRos
25
0
I apologize right now for the fact that I have no idea how to use LaTeX

I can't figure out if the following alternating series is convergent or not:

Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity

the root test is not applicable, A(n+1)>An, and the ratio test gives me Limit=1, so I have no comclusive evidence either way. Even Maple 10 couldn't give me an answer.
I have a test tomorrow. HELP!
 
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  • #2
Did you try the alternating series test?
 
  • #3
Just take a look at the limiting terms in that series. As n goes to infinity the absolute value of the terms goes to two, so alternating or not there is no way it can converge.
 
  • #4
BYW. I'm petty sure you can show that the above series is bounded, that is that it doesn't creep off to +/- infinity, but it definitely doesn't converge to a well defined limit point.
 
  • #5
You are aware, are you not, that is an does not go to zero, then the series[itex]\Sigma a_n[/itex] cannot converge? That's normally the very first "series" property you learn!
 

FAQ: Convergent or Divergent? Alternating Series Help for Tomorrow's Test

What is an alternating series?

An alternating series is a type of mathematical series in which the signs of the terms alternate between positive and negative. These series are typically used to approximate the sum of an infinite series.

What is the alternating series test?

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

How do you find the sum of an alternating series?

To find the sum of an alternating series, you can use the formula S = a - b + c - d + ..., where a, b, c, d, etc. are the terms in the series. This formula only works if the series satisfies the conditions of the alternating series test.

What is the remainder theorem for alternating series?

The remainder theorem for alternating series states that the error in using the alternating series approximation to approximate the sum of a series is less than or equal to the absolute value of the first neglected term.

What are some common examples of alternating series?

Some common examples of alternating series include the Leibniz series for pi, the alternating harmonic series, and the alternating geometric series. These series are often used in mathematics and physics to approximate values and functions.

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