Now prove that an is bounded above. This, too, is subtle.• The Binomial Theorem tells us thatп(п - 1)-x² +2!п(n - 1)(п — 2)(1+ x)" = 1+ nx ++...3!п(n - 1) -- (п - (п — 1))(п — 1)!n!..Deduce thatn(1+)"11<1+1+2!(n – 1)!3!n!• Show thatk!for any positive integer k.2k• Deduce thatn(1+ :)<1+52kk=0and that the partial sum is < 2. Thus an < 3 for all n.The Monotone Convergence Theorem tells us that the sequence converges to a limit< 3. In fact, that limit is e.

In this question, we prove the sequence an=(1+1n)n is convergent by given porcedure. i.e.…

Answered: prove that an is bounded above.

Math

Advanced Math

prove that an is bounded above.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 35E

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Question

Now prove that an is bounded above. This, too, is subtle.
• The Binomial Theorem tells us that
п(п - 1)
-x² +
2!
п(n - 1)(п — 2)
(1+ x)" = 1+ nx +
+...
3!
п(n - 1) -- (п - (п — 1))
(п — 1)!
n!
..
Deduce that
n
(1+)"
1
1
<1+1+
2!
(n – 1)!
3!
n!
• Show that
k!
for any positive integer k.
2k
• Deduce that
n
(1+ :)
<1+5
2k
k=0
and that the partial sum is < 2. Thus an < 3 for all n.
The Monotone Convergence Theorem tells us that the sequence converges to a limit
< 3. In fact, that limit is e.

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