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Convergence of this sequence .

  • Thread starter bs vasanth
  • Start date
  • #1
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Homework Statement


find the limit n[itex]\rightarrow[/itex]∞ of 10n/ n!


Homework Equations


L hospital rule


The Attempt at a Solution


took log and separated the num and denom as:
n ln10-ln(n!)
n ln10-n ln(n)+n
1/n ( ln10 - ln(n)+1)
now i applied l hospital rule
then i got lim n[itex]\rightarrow[/itex]∞ as 0.So the actual answer is 1. (e0)
I just want to know if the approach is correct.
 

Answers and Replies

  • #2
954
117
n ln10-n ln(n)+n
1/n ( ln10 - ln(n)+1)
I'm afraid I don't see how you transition between these two steps.
Also, intuitively the factorial function should dominate and the limit should be zero.
 
  • #3
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I'm afraid I don't see how you transition between these two steps.
Also, intuitively the factorial function should dominate and the limit should be zero.
my bad ,in my desperate attempt to get a ratio, I did that stupid thing. How do we solve it beyond intuition?
 
  • #4
954
117
From n ln10-n ln(n)+n, you can combine all the terms into a single expression. Taking the limit, you will find that the expression tends to minus infinity. The limit of the original expression is then 0.
 
  • #5
16
1
n*ln(10e/n) this is what i am getting , and it is not -infinity .
 
  • #6
Dick
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n*ln(10e/n) this is what i am getting , and it is not -infinity .
Yes, the limit of that is -infinity. n goes to +infinity. What does ln(10e/n) do?
 
  • #7
16
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Yes, the limit of that is -infinity. n goes to +infinity. What does ln(10e/n) do?
I took n out , then
n( ln10-lnn +1 )
n(ln10-lnn+lne)
n ln(10e/n)
i don't know what to do after this.
 
  • #8
16
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Are you saying that ln0 is taken as -inf, but there is also n before that right which becomes +inf.
 
  • #9
HallsofIvy
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The ratio test gives: [itex]a_{n+1}/a_n= (10^{n+1}/(n+ 1)!)(n!/10^n)= (10^{n+1})/10^n)(n!/(n+1)!)= 10/(n+1)[/itex] goes to 0 as n goes to 0. Strictly speaking, the ratio test shows that [tex]\sum a_n[/tex] converges but if that is so, then, of course, the sequence [itex]\{a_n\}[/itex] converges to 0.

(The converse is not necessarily true. For example, [itex]\{1/n\}[/itex] clearly converges to 0 but the sum [tex]\sum 1/n[/tex] does NOT converge so the ratio test does not work.)
 
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  • #10
Office_Shredder
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Are you saying that ln0 is taken as -inf, but there is also n before that right which becomes +inf.
n*ln(10e/n) if n is really big is a really big positive number multiplied by a really big negative number. What is that going to be?
 
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  • #11
16
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n*ln(10e/n) if n is really big is a really big positive number multiplied by a really big negative number. What is that going to be?
Now i get it , it is going to really big negative number, so -inf.
therefor the answer for the original question is 0.
thankyou everyone.
 

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