Converting Series for Comparison Test

Thread starter uzman1243
Start date Jun 3, 2014
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Comparison Comparison test Series Test

In summary, the comparison test for series is a method used to determine the convergence or divergence of an infinite series by comparing it to a known series. It works by comparing the terms of the given series to the terms of the known series. The difference between the comparison test and the limit comparison test is that the latter compares the ratio of the terms. The comparison test can only be used for series with positive terms and has limitations in finding the sum and requiring a known convergent or divergent series for comparison.

Jun 3, 2014

uzman1243

I'm trying to find if this series converges or diverges using the comparison test:

attachment.php?attachmentid=70324&stc=1&d=1401850854.png

and the answer goes:

attachment.php?attachmentid=70325&stc=1&d=1401850854.png

My problem is, I am not sure how to go from 1/2^(n+1) to 1/2(1/2)^n.

can you please explain that to me

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Jun 3, 2014

Matterwave

Science Advisor

Gold Member

$$\frac{1}{2^{n+1}}=\frac{1}{2\cdot 2^n}=\frac{1}{2}\frac{1}{2^n}=\frac{1}{2}\left(\frac{1}{2}\right)^n$$

Related to Converting Series for Comparison Test

1. What is the comparison test for series?

The comparison test for series is a method used to determine the convergence or divergence of an infinite series. It involves comparing the given series to a known series with known convergence or divergence behavior.

2. How does the comparison test work?

The comparison test works by comparing the terms of the given series to the terms of a known series. If the terms of the given series are smaller than the terms of the known series, and the known series converges, then the given series also converges. If the terms of the given series are larger than the terms of the known series, and the known series diverges, then the given series also diverges.

3. What is the difference between the comparison test and the limit comparison test?

The comparison test and the limit comparison test are two different methods used to determine the convergence or divergence of a series. The comparison test compares the terms of the given series to the terms of a known series, while the limit comparison test compares the ratio of the terms of the given series to the terms of a known series.

4. Can the comparison test be used for all series?

No, the comparison test can only be used for series with positive terms. It cannot be used for series with negative terms or alternating series.

5. Are there any limitations to the comparison test?

Yes, there are some limitations to the comparison test. It can only be used to determine the convergence or divergence of a series, it cannot be used to find the actual sum of a series. Additionally, the comparison test can only give a conclusive result if the known series used for comparison is known to converge or diverge.