(a) Let n be a positive integer greater than 2. Show that 1 1 +...+ 1 < In n <1+ 2 1 + 1 2 3 n-1 (Hint: compute the lower and upper sum of given the partition P = {1,2, 3, ...,n}.) (b) Use the result above to show the divergence of harmonic series, i.e. the sum 1. 1 1++ 1 1 +-+:+ 3 4 is infinite. (Hint: it suffices to show given any M > 0, there exists a positive integer n such that 1+++ .+ > M.)
(a) Let n be a positive integer greater than 2. Show that 1 1 +...+ 1 < In n <1+ 2 1 + 1 2 3 n-1 (Hint: compute the lower and upper sum of given the partition P = {1,2, 3, ...,n}.) (b) Use the result above to show the divergence of harmonic series, i.e. the sum 1. 1 1++ 1 1 +-+:+ 3 4 is infinite. (Hint: it suffices to show given any M > 0, there exists a positive integer n such that 1+++ .+ > M.)
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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Transcribed Image Text:(a) Let n be a positive integer greater than 2. Show that
1. 1
3
1
<1+
2
1
1
+ ...+
3
n-1
(Hint: compute the lower and upper sum of given the partition P = {1,2, 3, ...,n}.)
%3D
(b) Use the result above to show the divergence of harmonic series, i.e. the sum
1
1
1
+-+
2
4
1
1+
is infinite. (Hint: it suffices to show given any M > 0, there exists a positive integer n such
that 1+++.+ > M.)
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