Estimating error using comparison/limit comparison test

In summary, there is confusion about the proper formula for estimating error in a series approximation. Some sources use n for the p-series and n+1 for the geometric series, while others use n+1 for both. It is recommended to draw a diagram to determine the proper index to use for the integration area.
  • #1
motornoob101
45
0
I am confused about the right formula for this. Is it

[itex] R_{n} \leq T_{n} \leq \int_{n+1}^{\infty} [/itex] or [itex]R_{n} \leq T_{n} \leq\int_{n}^{\infty} [/itex]?

Say for example, I want to estimate the error in using the sum of the first 10 terms to approximate the sum of the series.

The textbook seems to use both methods (Use n for the p-series and n+1 for the geometric series)

(as seen in the examples here)

http://p3t3rl1.googlepages.com/textsolutionconfused.jpg

and I am quite confused which it is the right way. I am wondering if someone could clear this up for me. Thanks!
 
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  • #2
It is always best to draw a picture, not an exact one, but one that contains the crucial rectangles. The index depends on how the integration area is split into equidistant parts and whether you start counting by ##0## or by ##1##.
 

What is the comparison/limit comparison test?

The comparison/limit comparison test is a method used to estimate the error in a series or sequence by comparing it to a known series or sequence with known convergence properties.

How does the comparison/limit comparison test work?

The test works by comparing the terms of the given series or sequence to the terms of a known series or sequence with known convergence properties. If the given series or sequence has terms that are smaller than the corresponding terms in the known series or sequence, then it can be concluded that the given series or sequence also converges.

When should the comparison/limit comparison test be used?

This test should be used when the given series or sequence does not have a clear convergence or divergence pattern, and when it is difficult to use other tests such as the ratio test or the integral test.

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

The comparison test compares the given series or sequence to a known series or sequence, while the limit comparison test compares the ratio of the terms of the given series or sequence to the terms of the known series or sequence. The limit comparison test is generally considered to be more flexible and powerful compared to the comparison test.

What are the limitations of the comparison/limit comparison test?

The test can only be used to estimate the error in a series or sequence, and it cannot determine the exact value of the error. Additionally, the test may not always be applicable and may not provide accurate results for all series or sequences.

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