3. [Example 4.1.7 revisited] Let (bn)n≥1 be a sequence of real numbers such that lim bn = LE R. Define the sequence (an)n>1 by n-x an = bn -bn+1, n ≥ 1. n ak. (a) Simplify the nth partial sum Sn = Σ k=1 (b) Use (a) to prove that the (telescoping) series an is convergent. n>1 Find the sum of the series Σan. n=1 (c) Use (b) to prove that the series n>1 vergent and find the sum of the series. 1 n(n+1) m ((n ++1)n) is con-
3. [Example 4.1.7 revisited] Let (bn)n≥1 be a sequence of real numbers such that lim bn = LE R. Define the sequence (an)n>1 by n-x an = bn -bn+1, n ≥ 1. n ak. (a) Simplify the nth partial sum Sn = Σ k=1 (b) Use (a) to prove that the (telescoping) series an is convergent. n>1 Find the sum of the series Σan. n=1 (c) Use (b) to prove that the series n>1 vergent and find the sum of the series. 1 n(n+1) m ((n ++1)n) is con-
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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![3. [Example 4.1.7 revisited] Let (bn)n>1 be a sequence of real numbers
such that lim bn = LE R. Define the sequence (an)n>1 by
84x
an =
bn-bn+1, n ≥ 1.
n
(a) Simplify the nth partial sum Sn = Σ ak.
k=1
(b) Use (a) to prove that the (telescoping) series an is convergent.
n>1
Find the sum of the series Σ an.
n=1
(c) Use (b) to prove that the series
n>1
vergent and find the sum of the series.
1
n(n+1)
In
nn+1
(n+1) n
is con-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a71a13f-73d5-49c2-a029-9881c2af7703%2Fbc835590-fd77-4ce7-bc5b-1f75477d66ef%2Fhirkdjb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. [Example 4.1.7 revisited] Let (bn)n>1 be a sequence of real numbers
such that lim bn = LE R. Define the sequence (an)n>1 by
84x
an =
bn-bn+1, n ≥ 1.
n
(a) Simplify the nth partial sum Sn = Σ ak.
k=1
(b) Use (a) to prove that the (telescoping) series an is convergent.
n>1
Find the sum of the series Σ an.
n=1
(c) Use (b) to prove that the series
n>1
vergent and find the sum of the series.
1
n(n+1)
In
nn+1
(n+1) n
is con-
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