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Unformatted text preview: Ratio and Root Tests
John E. Gilbert, Heather Van Ligten, and Benni Goetz
Properties of geometric series enable us to test whether other series converge or diverge - even if they’re not geometric series! Root Test: an inﬁnite series
n an • • • Converges Absolutely when lim |an |1/n < 1,
n→∞ Diverges when lim |an |1/n > 1,
n→∞ if lim |an |1/n = 1, the test is inconclusive, it tells us
n→∞ nothing. Example 1: does the series
∞ an = (−1)n−1
n=1 n 2n + 1 n 2n + 1 n . For then |an |1/n = n 2n + 1
n 1/n converge absolutely ? Solution: the Root Test works well here because of the n -power exponent in
th = n 1 → 2n + 1 2 as n → ∞. So the series converges absolutely by the Root Test. What’s the connection with geometric series? Well,
n→∞ lim |an |1/n = L =⇒ |an | ≈ Ln for all large n, so
n |an | ≈
n Ln . But we know that the geometric series more care, this establishes the Root test.
n Ln converges when L < 1 and diverges when L > 1. With There’s another test similar to the Root Test, but one which often works well. It’s connection with the Geometric series is much the same as that for the Root Test, so we’ll ignore the details. Ratio Test: an inﬁnite series
n an an+1 an < 1, • • • Converges Absolutely when lim an+1 an n→∞ Diverges when lim an+1 an n→∞ > 1. If lim n→∞ = 1, the test is inconclusive, it tells us nothing. Example 2: does the series
∞ For then an+1 3(n + 1)2 n! = = an (n + 1)! 3n2 in which case an+1 an the Ratio Test. = 1+ 1 n
n=1 2 n−1 3 n n! n+1 n 2 n! , (n + 1) n! converge absolutely ? Solution: the Ratio Test works well here because of the n! factorial term in an = (−1)n−1 3n2 . n! 1 →0 n+1 as n → ∞. So the series converges absolutely by Neither the Root or Ratio Test, however, works well on Example 3: is the series
n=1 4n 1 + n3 the Basic Comparison Test together with the 1 p-series shows that 2 nn
∞ n=1 ∞ absolutely convergent, conditionally convergent or divergent ? Solution: since 0< 4n 4 <2 3 1+n n converges. So absolutely. 4n 1 + n3 4n converges 1 + n3 (−1)n−1
n=1 INFINITE SERIES SUMMARY
1. Convergence, Divergence: 2. Geometric series: 3. Harmonic series, p-series: 4. Integral Test: 5. Basic Comparison Test: 6. Limit Comparison Test: 7. Divergence Test: 8. Alternating Series Test: 9. Absolute, Conditional Convergence: 10. Root Test: 11. Ratio Test: ...
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This note was uploaded on 03/01/2011 for the course PHYS 317K taught by Professor Turner during the Spring '11 term at University of Texas at Austin.
- Spring '11