Choosing Between the Ordinary & Limit Comparison Test

In summary, the conversation is about choosing between the ordinary comparison test and the limit comparison test. The person decided to use the limit comparison test for two specific problems. They mention that when using the limit comparison test, it is important to look for the largest degree of terms in the numerator and denominator and divide through, which in this case is 3/n^2. They ask why 3/n^2 is the largest term and not 3/n^3, to which the expert responds that it is because the relevant term in the numerator is 3*n, making it 3*n/n^3=3/n^2. The person also asks what series does 3/n^2 belong to, to which the expert does
  • #1
rcmango
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Homework Statement



First questions is: How to choose between using the ordinary comparison test, or using the limit comparison test?



Homework Equations





The Attempt at a Solution



then, for these two problems below, i decide to use the limit comparison test:

SUM n_infinity (3n - 2)/(n^3 - 2n^2 + 11) then its said to look for the largest degree of terms in the numerator and denominator and divide through which is to divide by: 3/n^2

but why is 3/n^2 the largest term, when there is a n^3 in there, why not 3/n^3 ?

=======================

also, i see that 1/n belongs to the harmonic series, but what series does 3/n^2 belong to?

thankyou.
 
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  • #2
Because the relevant term in the numerator is 3*n. 3*n/n^3=3/n^2.
 

What is the Ordinary Comparison Test and how is it used in choosing between tests?

The Ordinary Comparison Test, also known as the Direct Comparison Test, is a method used to determine the convergence or divergence of a series by comparing it to a known series with known convergence or divergence. It compares the given series to a larger and smaller series to determine if the given series behaves similarly to the known series.

How is the Limit Comparison Test different from the Ordinary Comparison Test?

The Limit Comparison Test is similar to the Ordinary Comparison Test in that it also compares the given series to a known series. However, instead of comparing the series term by term, it compares the ratio of the terms as n approaches infinity. This allows for a more accurate determination of convergence or divergence.

When should I use the Ordinary Comparison Test versus the Limit Comparison Test?

The choice between the Ordinary and Limit Comparison Tests depends on the given series. The Ordinary Comparison Test is useful for series with non-negative terms, while the Limit Comparison Test is more versatile and can be used for a wider range of series. Additionally, the Limit Comparison Test is more accurate and efficient for determining convergence or divergence.

Are there any limitations to using the Ordinary and Limit Comparison Tests?

While these tests are useful for many series, they do have their limitations. The Ordinary Comparison Test cannot be used for series with negative terms and the Limit Comparison Test may not work for series with alternating signs. Additionally, both tests rely on having a known series to compare to, which may not always be available.

What are some tips for choosing between the Ordinary and Limit Comparison Tests?

When faced with choosing between these two tests, it is important to consider the given series and its terms. If the series has non-negative terms, the Ordinary Comparison Test may be a good choice. If the series has alternating signs or if the terms are complicated, the Limit Comparison Test may be more useful. It is also helpful to have a good understanding of the convergence and divergence of common series, as this can aid in choosing the appropriate test.

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