A sequence (In) is defined by Io = 1 and for every n E N* , In = Integral from 0→1 [dx / ( 1 + x^2 )^n ]
(a) Justify that In is welled defined and determine its sign.
(b) Show that In is a decreasing sequence.
1
Expert's answer
2021-02-02T05:27:53-0500
As ∀n∈N,(x2+1)n1 is a continuous function on [0;1] and [0;1] is a compact interval, ∀n∈N,In is well defined. In addition, as (1+x2)n1≥0, the integral In≥0.
As 1+x21≤1, (1+x2)n+11≤(1+x2)n1 and thus In+1≤In, the sequence is non-increasing. To show that it is strictly decreasing, it is enough to decompose this integral into two parts : In=∫01/2(x2+1)n1dx+∫1/21(x2+1)n1dx, the first part is non-increasing due to the same inequality as before, in the second part x2+11≤54<1 and thus (1+x2)n+11<(1+x2)n1,x∈[1/2;1] and so the second part is strictly decreasing. The sum of a non-increasing and a strictly secreasing sequences is a strictly decreasing sequence and thus In is decreasing.
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