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Using the ratio test

For each of the following series, use the ratio test to determine whether the series converges or diverges.

  1. n = 1 2 n n !
  2. n = 1 n n n ! n = 1 ( −1 ) n ( n ! ) 2 ( 2 n ) !
  3. n = 1 ( −1 ) n ( n ! ) 2 ( 2 n ) !
  1. From the ratio test, we can see that
    ρ = lim n 2 n + 1 / ( n + 1 ) ! 2 n / n ! = lim n 2 n + 1 ( n + 1 ) ! · n ! 2 n .

    Since ( n + 1 ) ! = ( n + 1 ) · n ! ,
    ρ = lim n 2 n + 1 = 0 .

    Since ρ < 1 , the series converges.
  2. We can see that
    ρ = lim n ( n + 1 ) n + 1 / ( n + 1 ) ! n n / n ! = lim n ( n + 1 ) n + 1 ( n + 1 ) ! · n ! n n = lim n ( n + 1 n ) n = lim n ( 1 + 1 n ) n = e .

    Since ρ > 1 , the series diverges.
  3. Since
    | ( −1 ) n + 1 ( ( n + 1 ) ! ) 2 / ( 2 ( n + 1 ) ) ! ( −1 ) n ( n ! ) 2 / ( 2 n ) ! | = ( n + 1 ) ! ( n + 1 ) ! ( 2 n + 2 ) ! · ( 2 n ) ! n ! n ! = ( n + 1 ) ( n + 1 ) ( 2 n + 2 ) ( 2 n + 1 )

    we see that
    ρ = lim n ( n + 1 ) ( n + 1 ) ( 2 n + 2 ) ( 2 n + 1 ) = 1 4 .

    Since ρ < 1 , the series converges.
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Use the ratio test to determine whether the series n = 1 n 3 3 n converges or diverges.

The series converges.

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Root test

The approach of the root test    is similar to that of the ratio test. Consider a series n = 1 a n such that lim n | a n | n = ρ for some real number ρ . Then for N sufficiently large, | a N | ρ N . Therefore, we can approximate n = N | a n | by writing

| a N | + | a N + 1 | + | a N + 2 | + ρ N + ρ N + 1 + ρ N + 2 + .

The expression on the right-hand side is a geometric series. As in the ratio test, the series n = 1 a n converges absolutely if 0 ρ < 1 and the series diverges if ρ 1 . If ρ = 1 , the test does not provide any information. For example, for any p -series, n = 1 1 / n p , we see that

ρ = lim n | 1 n p | n = lim n 1 n p / n .

To evaluate this limit, we use the natural logarithm function. Doing so, we see that

ln ρ = ln ( lim n 1 n p / n ) = lim n ln ( 1 n ) p / n = lim n p n · ln ( 1 n ) = lim n p ln ( 1 / n ) n .

Using L’Hôpital’s rule, it follows that ln ρ = 0 , and therefore ρ = 1 for all p . However, we know that the p -series only converges if p > 1 and diverges if p < 1 .

Root test

Consider the series n = 1 a n . Let

ρ = lim n | a n | n .
  1. If 0 ρ < 1 , then n = 1 a n converges absolutely.
  2. If ρ > 1 or ρ = , then n = 1 a n diverges.
  3. If ρ = 1 , the test does not provide any information.

The root test is useful for series whose terms involve exponentials. In particular, for a series whose terms a n satisfy | a n | = b n n , then | a n | n = b n and we need only evaluate lim n b n .

Using the root test

For each of the following series, use the root test to determine whether the series converges or diverges.

  1. n = 1 ( n 2 + 3 n ) n ( 4 n 2 + 5 ) n
  2. n = 1 n n ( ln ( n ) ) n
  1. To apply the root test, we compute
    ρ = lim n ( n 2 + 3 n ) n / ( 4 n 2 + 5 ) n n = lim n n 2 + 3 n 4 n 2 + 5 = 1 4 .

    Since ρ < 1 , the series converges absolutely.
  2. We have
    ρ = lim n n n / ( ln n ) n n = lim n n ln n = by L’Hôpital’s rule .

    Since ρ = , the series diverges.
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Use the root test to determine whether the series n = 1 1 / n n converges or diverges.

The series converges.

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Choosing a convergence test

At this point, we have a long list of convergence tests. However, not all tests can be used for all series. When given a series, we must determine which test is the best to use. Here is a strategy for finding the best test to apply.

Problem-solving strategy: choosing a convergence test for a series

Consider a series n = 1 a n . In the steps below, we outline a strategy for determining whether the series converges.

  1. Is n = 1 a n a familiar series? For example, is it the harmonic series (which diverges) or the alternating harmonic series (which converges)? Is it a p series or geometric series? If so, check the power p or the ratio r to determine if the series converges.
  2. Is it an alternating series? Are we interested in absolute convergence or just convergence? If we are just interested in whether the series converges, apply the alternating series test. If we are interested in absolute convergence, proceed to step 3 , considering the series of absolute values n = 1 | a n | .
  3. Is the series similar to a p series or geometric series? If so, try the comparison test or limit comparison test.
  4. Do the terms in the series contain a factorial or power? If the terms are powers such that a n = b n n , try the root test first. Otherwise, try the ratio test first.
  5. Use the divergence test. If this test does not provide any information, try the integral test.
Practice Key Terms 2

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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