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(a) Let n be a positive integer greater than 2. Show that 1 1 + + 3 1 + 1 1 < Inn <1+ + 2 3 1 + .. п — 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+ 2 1 1 1 += +... 3 is infinite. (Hint: it suffices to show given any M > 0, there exists n such that 1++ + ..+> M.)

(a) Let n be a positive integer greater than 2. Show that 1 1 + + 3 1 + 1 1 < Inn <1+ + 2 3 1 + .. п — 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+ 2 1 1 1 += +... 3 is infinite. (Hint: it suffices to show given any M > 0, there exists n such that 1++ + ..+> M.)

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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Problem 2. (a) Let n be a positive integer greater than 2. Show that 1 1 1 1 1 1 < Inn <1+ + 3 n 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+ 2 1 + 5 3 4 is infinite. (Hint: it suffices to show given any M > 0, there exists n such that 1+ ++ +> M.)
Transcribed Image Text:Problem 2. (a) Let n be a positive integer greater than 2. Show that 1 1 1 1 1 1 < Inn <1+ + 3 n 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+ 2 1 + 5 3 4 is infinite. (Hint: it suffices to show given any M > 0, there exists n such that 1+ ++ +> M.)
Expert Solution
Step 1

(a)

Given n be a positive integer greater than 2 

show that  12+13+......+1n<lnn<1+12+13+......+1n-1

here we have to compute the upper sum and lower sum of 1x given partition P={1, 2, 3,.....,n}

Now by the definition the lower and upper Riemann sum L(f, P) ,U(f, P) respectively defined as

For L(f, P)= ∑k=1nmi(xi-xi-1)  where mi=inff(x):x∈[xi-1,xi

and U(f, P) =∑k=1nmi(xi-xi-1)   where mi=sup{f(x):x∈[xi-1,xi}

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