Answer to Question #184539 in Real Analysis for Leonard

Question #184539
  1. Use mathematical induction to show that n! ≥ 2(n-1) for all n ≥ 1.
  2. Use (1) and the definition of a Cauchy sequence to show that Sn = ( 1+ 1/2! + 1/3! + ⋯ 1/n!) is Cauchy sequence.
1
Expert's answer
2021-05-07T08:50:02-0400


(i)


n! ≥ 2(n-1)


For P(2) = "2! \t\\geq 2(2-1)"

2=2


for P(3)= "3! \t\\geq 2(3-1)"

6>4


now for

P(k)


The inductive hypothesis is

"\\boxed{k! \t\\geq 2(k-1)}"



(ii)

by using


n! ≥ 2(n-1)


and using definition of a Cauchy sequence


"|a_{n+p}-a_{n}|= \\frac{1}{(n+1)!}+\\frac{1}{(n+2)!}+..........+\\frac{1}{(n+p)!}"


"\\leq \\frac{1}{n(n+1)}+\\frac{1}{(n+1)(n+2)}+...........+\\frac{1}{(n+p-1)(n+2)}"

"=\\frac{1}{n}-\\frac{1}{n+p}"


"\\leq\\frac{1}{n}"

hence proved

"\\boxed{S_n = ( 1+\\frac{1}{2!}+\\frac{1}{3!}..................+\\frac{1}{n!} \n}"



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