Convergence & Sum of Alternating Series | Homework Help

In summary, the given series is an alternating geometric series that is convergent. The sum of the series can be approximated using a comparison test and the accuracy can be improved by increasing the number of terms used in the calculation.
  • #1
rcmango
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Homework Statement



Determine wheter the series is convergent or divergent. If it convergent, approximate the sum of the series correct to four decimal places.

heres the equation: http://img251.imageshack.us/img251/2261/46755781zg9.png [Broken]

Homework Equations





The Attempt at a Solution



This appears to be an alternating geometric series,

Would it be okay to move the exponent k over everything? in other words: ( (-1)/k) )^k

So then it looks a lot like a geometric series, so then It converges by the rules of an alernating series, it is decreasing and it is approaching zero.

So then to find its sum, i would do so by geometric series right?

first term would be starting at k = 2, so: 1/2?

then use 1/2 divided by 1 -r

Am i on the right track? what is r? is it also, 1/2?
 
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  • #2
Yes you can use k as the exponent of the whole because of the distributive property of exponentiation.
 
  • #3
how about the rest of what I'm doing here, this was my best hypothesis to approach the problem. I need help with the common ratio. I'm not sure what to use if its k^k ?
 
  • #4
rcmango said:
Determine wheter the series is convergent or divergent. If it convergent, approximate the sum of the series correct to four decimal places.

heres the equation: http://img251.imageshack.us/img251/2261/46755781zg9.png [Broken]

This appears to be an alternating geometric series...

It isn't a geometric series because such series has a constant ratio between successive terms. However, that gives you a clue to the proof of its convergence. (Try a comparison test.) As for the estimate of the sum, do they want an analytical proof of some sort or just something carried out on a calculator (how many terms do you need to get to a precision of 10^-4 ?)
 
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1. What is convergence and sum of alternating series?

Convergence refers to the behavior of a series where the terms are added together to form a sum. In the case of an alternating series, the terms alternate between positive and negative, and the sum is determined by the limit of the partial sums of the series.

2. How do you determine if an alternating series converges or diverges?

To determine convergence or divergence of an alternating series, there are several tests that can be used such as the alternating series test, ratio test, or root test. These tests involve evaluating the limit of the series and comparing it to known values to determine convergence or divergence.

3. What is the alternating series test?

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

4. Can an alternating series converge to a specific value?

Yes, an alternating series can converge to a specific value if it satisfies the conditions of the alternating series test. If the series satisfies these conditions, then the limit of the series will be equal to the sum of the series.

5. How can I use the alternating series test to determine convergence?

To use the alternating series test, you will need to evaluate the limit of the series and compare it to known values. If the limit is equal to zero or a known value such as pi or e, then the series will converge. If the limit is not equal to a known value, then further tests may be needed to determine convergence or divergence.

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