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Let {an}o be a sequence with positive terms. (a) Prove that if lim van= ∞, then La, diverges. (b) Prove that if lim van= L> 1, then Can diverges. Hint. Start by using the definition of that limit to show that there is an r > 1 such that, for sufficiently large n, Van >r. (c) Prove that if lim yan= L< 1, then Ca, converges. %3D

Let {an}o be a sequence with positive terms. (a) Prove that if lim van= ∞, then La, diverges. (b) Prove that if lim van= L> 1, then Can diverges. Hint. Start by using the definition of that limit to show that there is an r > 1 such that, for sufficiently large n, Van >r. (c) Prove that if lim yan= L< 1, then Ca, converges. %3D

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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One way to to characterize a geometric series is that the ratio between successive terms is constant. That is, b, = r" for all n EN (i.e., E bn is a goometric series) if and only if bo =1 and nil = r for all n E N. bn We also know that a geometric series r" converges if and only if |r| < 1. Given some series an, if we know that lim "nt1 = L < 1, then eventually is bounded above by some number less than 1, and in turn a, is eventually bounded above by a convergent geometric scries. Then a,n converges by the BCT with that gcometric scries. In this exercise, you will prove another convergence test called the Root Test. This test is based on a different way of characterizing a geometric series: bn = r" for all n EN (i.c., E bn is a geometric series) if and only if Vn =r for all nɛ N. Let {an}o be a sequence with positive terms. (a) Prove that if lim van = 0, then Ca, diverges. (b) Prove that if lim van = L> 1, then an diverges. Hint. Start by using the definition of that limit to show that there is an r >1 such that, for sufficiently large n, van > r. (c) Prove that if lim ya, = L< 1, then (a, converges. Hint. Start by using the definition of that limit to show that there is an r <1 such that, for sufficiently large n, yan <r. (d) Show with two examples that if lim yan = 1, we cannot conclude anything about the convergence of Ean. (e) Conclude from the first two parts that if {b„}o is any sequence (not necessarily with positive terms), then: • If lim Vb = L > 1 or = xo, then b, diverges. • If lim Vbn = L< 1, then £b, converges absolutely. This is called the Root Test.
Transcribed Image Text:One way to to characterize a geometric series is that the ratio between successive terms is constant. That is, b, = r" for all n EN (i.e., E bn is a goometric series) if and only if bo =1 and nil = r for all n E N. bn We also know that a geometric series r" converges if and only if |r| < 1. Given some series an, if we know that lim "nt1 = L < 1, then eventually is bounded above by some number less than 1, and in turn a, is eventually bounded above by a convergent geometric scries. Then a,n converges by the BCT with that gcometric scries. In this exercise, you will prove another convergence test called the Root Test. This test is based on a different way of characterizing a geometric series: bn = r" for all n EN (i.c., E bn is a geometric series) if and only if Vn =r for all nɛ N. Let {an}o be a sequence with positive terms. (a) Prove that if lim van = 0, then Ca, diverges. (b) Prove that if lim van = L> 1, then an diverges. Hint. Start by using the definition of that limit to show that there is an r >1 such that, for sufficiently large n, van > r. (c) Prove that if lim ya, = L< 1, then (a, converges. Hint. Start by using the definition of that limit to show that there is an r <1 such that, for sufficiently large n, yan <r. (d) Show with two examples that if lim yan = 1, we cannot conclude anything about the convergence of Ean. (e) Conclude from the first two parts that if {b„}o is any sequence (not necessarily with positive terms), then: • If lim Vb = L > 1 or = xo, then b, diverges. • If lim Vbn = L< 1, then £b, converges absolutely. This is called the Root Test.
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