QUESTION 4

Question 4. More on harmonic series {6 marks} [a] The harmonic series diverges very slowly Prove that the sum of the first 105 terms is less than 15 and that the sum of the rst 109 terms is less than 22. {You should not actually calculate these sums!) [2 marks] in [1:] Determine whelher the series E 10n1+ 1 is convergent or divergent. [2 marks] n=1 (c) Prove that the sequence dened by 1 1 1 1 t1+E++E+HI+E_IDE"[\"} has a limit. [2 marks] Hint. Show that in 2:- D and in :- t..+1 for all n and then applyr the monotone bounded sequence theorethprovemeseinequalitiesJetyourselfbeinspiredbytheargumentsmdpicturesmthe section on re integral oomparison test from the lectures and lecture notes. What does in represent pictorially