Integral test and ratio test on haromonic series.

Thread starter avman
Start date Aug 9, 2012
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Integral Integral test Ratio Ratio test Series Test

In summary, the integral test and ratio test are two methods used to determine the convergence or divergence of infinite series. The integral test involves comparing the series to an improper integral, while the ratio test involves taking the limit of the ratio between consecutive terms. While the results of these tests may not always agree, they can be used to confirm the convergence or divergence of a series. When applied to the harmonic series, the integral test shows that it diverges, while the ratio test shows that it is not absolutely convergent.

Aug 9, 2012

avman

So harmonic series diverges because of the integral test but if I try it on ratio test

= (1 / ( x+1 )) / (1 / x)
= x / (x + 1) and this is less than 1 so shouldn't it converge?

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Aug 9, 2012

Millennial

The limit as x goes to infinity is 1, so the ratio test is inconclusive. It does not tell us whether the harmonic series converges or diverges.

1. What is the integral test?

The integral test is a method used to determine the convergence or divergence of infinite series. It involves comparing the series to an improper integral and using the properties of integrals to determine its convergence.

2. How does the integral test work?

The integral test works by comparing the infinite series to an improper integral. If the integral converges, then the series also converges. Conversely, if the integral diverges, then the series also diverges.

3. What is the ratio test?

The ratio test is another method used to determine the convergence or divergence of infinite series. It involves taking the limit of the ratio between consecutive terms of the series and using this limit to determine its convergence or divergence.

4. How is the ratio test different from the integral test?

While both tests are used to determine the convergence or divergence of series, the ratio test relies on taking the limit of the ratio between terms, while the integral test involves comparing the series to an improper integral. The results of the two tests may not always agree, and it is important to use both methods to confirm the convergence or divergence of a series.

5. How do the integral test and ratio test apply to the harmonic series?

The harmonic series is a special case where the terms of the series are given by the reciprocals of positive integers. The integral test can be used to show that the harmonic series diverges, while the ratio test can be used to show that the series is not absolutely convergent. This means that while the series does not converge, it also does not converge when the absolute values of its terms are considered.