Convergence of the Alternating Series: Investigating (2/3)^n Summation

In summary, the conversation discusses the convergence of the alternating series (-1)^n+1 (2/3)^n and how to take the limit of (2/3)^n as n approaches infinity. It is mentioned that the limit of (2/3)^n is equal to 0 and this can be shown using the Geometric Series test. The conversation also mentions that this limit can be found because (2/3)^n is a geometric series with |2/3|<1.
  • #1
J.live
95
0

Homework Statement

summation --> (-1)^n+1 (2/3)^n (I don't know how to do the symbol for sum)

The Attempt at a Solution

I) lim n-->∞ (2/3)^n = limit does not exist ? It diverges ?


P.S I am not sure if this is true. Any explanation will be a great help. Thanks.
 
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  • #2
The limit
[tex] \lim_{n\rightarrow +\infty}{(2/3)^n} [/tex]
does exist. It equals zero. So no help there.

You can in fact show that
[tex] \sum_{n=0}^n{(2/3)^n} [/tex]
converges (so the original alternating series is absolutely convergent). Can you show this? HINT: it's a geometric progression and thus the exact limit can be found...
 
  • #3
Can you please explain how you applied n --> ∞ to (2/3)^n = (2/3) ^∞ How do i solve this?
 
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  • #4
You have without a doubt seen that

[tex] a^n\rightarrow 0 [/tex]

if |a|< 1. Haven't you?
 
  • #5
Yes, I am aware of the Geometric Series test. That was not my question.

My question is how did you derive to answer zero when you replaced n with ∞.

I am aware that (2/3)^n is a geometric series.I am having trouble with taking the limit of (2/3)^n.

i.e. when lim n-->∞ 1/n = 1/∞ = 0. Similarly, how do I take the limit n-->∞ in this case ?
 
  • #6
I wasnt talking about geometric series. I was talking about the sequence [tex] (a^n)_n [/tex] with a<1. Such a sequence always has

[tex] \lim_{n\rightarrow +\infty} {a^n} = 0 [/tex]

You must have seen this.
 
  • #7
Yes, I have seen this. Ah, makes sense. Thanks.
 
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Related to Convergence of the Alternating Series: Investigating (2/3)^n Summation

1. 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 0, then the series will converge.

2. How do you apply the Alternating Series Test?

To apply the Alternating Series Test, you must first check that the terms of the series alternate from positive to negative. Then, you must check that the terms decrease in absolute value and approach 0. If both of these conditions are met, then the series is convergent.

3. What is the difference between an alternating series and a regular series?

An alternating series is a series in which the terms alternate between positive and negative, while a regular series has terms that are all either positive or negative. The Alternating Series Test can only be applied to alternating series.

4. Can the Alternating Series Test be used to determine the exact value of a series?

No, the Alternating Series Test only tells us whether a series converges or diverges. It does not give us the exact value of the series.

5. Are there any other tests that can be used to determine the convergence of a series?

Yes, there are several other tests such as the Ratio Test, Root Test, and Limit Comparison Test that can also be used to determine the convergence or divergence of a series. It is important to use multiple tests to confirm the convergence or divergence of a series.

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