Consider the sequence {n!/n^n}: a) Use one of the tests for monotonicity to dete
ID: 2887954 • Letter: C
Question
Consider the sequence {n!/n^n}:
a) Use one of the tests for monotonicity to determine whether the given sequence is increasing, decreasing, or not monotonic.
b)Determine whether this sequence is convergent, if so find the limit.
c)Determine whether this sequence is bounded. Justify your answer.
*Please excuse notes taken on sheet*
4. Consider the sequence { n! (a) Use one of the tests for monotonicity to determine whether the given sequence is increasing decreasing, or not monotonic (b) Determine whether this sequence is convergent. If so, find the limit. (c) Determine whether this sequeres s bounded. Justify your answer.Explanation / Answer
(a)
an= n!/nn
an+1= (n+1)!/(n+1)n+1
an+1= (n+1)n!/((n+1)(n+1)n)
an+1= n!/(n+1)n
n!/(n+1)n< n!/nn
=>an+1<an
so sequence is decreasing
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(b)
lim[n->] an=lim[n->] (n!/nn)
lim[n->] an=lim[n->] (n*(n-1)*(n-2)*....*2*1/n*n*n*..*n*n)
lim[n->] an=lim[n->] 1*(1-(1/n))*(1-(2/n))*....*(2/n)*(1/n)
lim[n->] an= 1*(1-0)*(1-0)*....*(0)*(0)
lim[n->] an=0
limit is 0, sequence is convergent
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(c)
All he terms in this sequence are positive and so it is bounded below by zero
and above it is not bounded
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