11. (Bernoulli's Inequality) Show that for a natural number n and a nonnegative number b, (1+b)" ≥ 1+nb. (Hint: In the Binomial Formula, set a = 1.) 1829
11. (Bernoulli's Inequality) Show that for a natural number n and a nonnegative number b, (1+b)" ≥ 1+nb. (Hint: In the Binomial Formula, set a = 1.) 1829
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Solve number 11 please

Transcribed Image Text:20
DODA
ADVANCED CALCULUS
Then replace b by -b to obtain
Po
|la|-|b|| ≤la - bl.
8. Let a and b be numbers such that la- b) ≤ 1. Prove that |a| ≤ |b| + 1.
9. For a natural number n and any two nonnegative numbers a and b, use the Difference
of Powers Formula to prove that
a ≤b
if and only if a" ≤b".
10. For a natural number n and numbers a and b such that a b≥ 0, prove that
a"-b" ≥nb"-¹ (a - b).
11. (Bernoulli's Inequality) Show that for a natural number n and a nonnegative
number b,
(1+b)" ≥ 1+nb.
(Hint: In the Binomial Formula, set a = 1.)
12. Use the Principle of Mathematical Induction to provide a direct proof of Bernoulli's
Inequality for all b> -1, not just for the case where b≥ 0 which, as outlined ins
Exercise 11 follows from the Binomial Formula.
13. For a natural number n and a nonnegative number b show that
n(n-1) ².
2
(1+b)" ≥ 1+nb+
14. (Cauchy's Inequality) Using the fact that the square of a real number is nonnegativ
prove that for any numbers a and b,
by
is Sneg
-2 pos it pos
1
ab ≤ = (a² + b²).
15. Use Cauchy's Inequality to prove that if a 20 and b≥ 0, then
√ab ≤ = (a + b).
16. Use Cauchy's Inequality to show that for any numbers a and b and a nat
number n.
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