Question

please help with b) and c)

Thank you:)

(a) The harmonic series diverges very slowly. Prove that the sum of the first 106 terms is less than 15 and that the sum of the first 10° terms is less than 22. (You should not actually calculate these sums!) [2 marks] (b) Determine whether the series 10n +1 101 is convergent or divergent. [3 marks] (c) Consider the series whose terms are the reciprocals of the positive integers whose decimal represen- tation does not include the digit 0. For example, and are not terms in the series. Show that this series converges and that its sum is less than 100. [3 marks) Hint. Group the terms of the series in the following way. 1 (1+2+...+5)+(1+2+.. 1 1 1 1 + ... + = + + 12 +... 99 111 112 + ... + 999 Now estimate the series 81 +82 +83 +... by counting the number of terms in gn and observing that each term in 8n is less than 10-

Answer 1

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Here Commanly we use ratios test and Camparison test