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.
The expert did excellent work as usual and was extremely helpful for me.
"Assignmentexpert.com" has experienced experts and professional in the market. Thanks.
Comments