(Inx)* Given that n is a natural number and (In) is the sequence defined by In = a.) Show that (I,) is welled defined and positive.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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Exercise 1
Given that n is a natural number and (I,) is the sequence defined by I, =
e(Inx)"
%3D
a.) Show that (!,) is welled defined and positive.
b.) Show that 0 s S1.
c.) Show that (In) is a decreasing sequence.
d.) Deduce that (I,) is convergent and calculate the limit of.
e.) Show that I1 = -+ (n + 1)
f.) Hence, show further that , = 1-E
8.) Prove from the preceding questions that lim E-o = e.
Zk=0 k
n+00
Transcribed Image Text:Exercise 1 Given that n is a natural number and (I,) is the sequence defined by I, = e(Inx)" %3D a.) Show that (!,) is welled defined and positive. b.) Show that 0 s S1. c.) Show that (In) is a decreasing sequence. d.) Deduce that (I,) is convergent and calculate the limit of. e.) Show that I1 = -+ (n + 1) f.) Hence, show further that , = 1-E 8.) Prove from the preceding questions that lim E-o = e. Zk=0 k n+00
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