mmilton said:Homework Statement
Is the sequence {((-1)^n)/2n} convergent? If so, what is the limit?
Homework Equations
The Attempt at a Solution
I'm thinking that it is convergent by the alternating series test, but I am not certain. The limit part I'm not sure how to go about it. Is it simply the sum, or is that something completely different. Thanks so much for any hints or suggestions.
mmilton said:It is a sequence. So, does that preclude using the alternating series test (i.e. is that only used on series)?
mmilton said:It is a sequence. So, does that preclude using the alternating series test (i.e. is that only used on series)?
Ray Vickson said:(The series Ʃ(-1)^n/(2n) also converges, by the "alternating series test"---but that is a totally different question!)
Whovian said:I think it would be a bad idea to use the Alternating Series test, the limit of the series of the absolute value of this diverges.
Whovian said:Huh? I thought by the AST, we'd get that if Ʃ1/2n converges, then so does the original series. Which it doesn't. So we have to use some other test.
Whovian said:Huh? I thought by the AST, we'd get that if Ʃ1/2n converges, then so does the original series. Which it doesn't. So we have to use some other test.
LCKurtz said:First, the OP keeps claiming that he has a sequence problem, not a series. I admit, since he keeps using the term "sum" here and there, I'm not sure I believe him. Secondly, if it really is an alternating series, you need to re-look at what the AST says. The alternating series converging does not mean the series of absolute values does. Read RGV's explanation again.
Ray Vickson said:The alternating series test is used for SERIES---that is, for a series where the sequence of terms is alternating in sign. If you really mean the sequence {(-1)^n/(2n)}, then that converges because the terms --> 0 as n --> ∞. (The series Ʃ(-1)^n/(2n) also converges, by the "alternating series test"---but that is a totally different question!)
RGV
Mark44 said:I think that you are confused about what the alternating series test says.
The AST can be used to determine the convergence of this series:
$$\sum_{n = 1}^{\infty}\frac{(-1)^n}{2n} $$
The sequence {((-1)^n)/2n} is an alternating series that consists of positive and negative terms, where the denominator increases by 2 with each term and the numerator alternates between positive and negative 1.
A convergent sequence is a sequence of numbers that approaches a finite limit as the number of terms increases. In other words, as the terms of the sequence continue, the values get closer and closer to a specific number.
To determine if a sequence is convergent, we can use different tests such as the limit comparison test, the ratio test, or the root test. These tests involve evaluating the limit of the sequence and comparing it to known values to determine if it converges or diverges.
Yes, the sequence {((-1)^n)/2n} is convergent. It can be proven using the alternating series test, which states that if a series satisfies certain conditions, then it must converge. In this case, the sequence satisfies the conditions of alternating signs and the limit of the absolute value of the terms approaches 0 as n approaches infinity.
The limit of the sequence {((-1)^n)/2n} is 0. This can be seen by evaluating the limit of the absolute value of the terms, which approaches 0 as n approaches infinity. Therefore, we can conclude that the sequence converges to 0.