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4. Prove the following corollary to the root test: If p(n) = |an|/n converges to a limi then: • If L < 1, then the series converges absolutely. If L S 1 then the series diverges

4. Prove the following corollary to the root test: If p(n) = |an|/n converges to a limi then: • If L < 1, then the series converges absolutely. If L S 1 then the series diverges

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Prove the following corollary to the root test: If p(n) = |anl/" converges to a limit L, then: • If L < 1, then the series converges absolutely. • If L > 1 then the series diverges.
Transcribed Image Text:4. Prove the following corollary to the root test: If p(n) = |anl/" converges to a limit L, then: • If L < 1, then the series converges absolutely. • If L > 1 then the series diverges.
Theorem 3.9 (Root test, limit root test). Let an be a sequence of real numbers. Let p(n) = |an|1/". Then: (i) If there exists c < 1 and N >1 so that p(n) < c for all n 2 N, then the series an converges absolutely. (ii) If there exist arbitrarily large n so that p(n) > 1, then the series diverges. Consequently, if p(n) → L, then we obtain: • If L < 1, then the series converges absolutely. • If L > 1, then the series diverges. • If L = 1, then the test is inconclusive.
Transcribed Image Text:Theorem 3.9 (Root test, limit root test). Let an be a sequence of real numbers. Let p(n) = |an|1/". Then: (i) If there exists c < 1 and N >1 so that p(n) < c for all n 2 N, then the series an converges absolutely. (ii) If there exist arbitrarily large n so that p(n) > 1, then the series diverges. Consequently, if p(n) → L, then we obtain: • If L < 1, then the series converges absolutely. • If L > 1, then the series diverges. • If L = 1, then the test is inconclusive.
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