Proving lim 10^n/n!=0 Using Limit Theorems

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In summary, to prove that lim (10^n / n!) = 0 using limit theorems, we can use the fact that lim(s_n) = infinity implies lim(1/s_n) = 0, where s_n is a sequence. By expanding n! as n(n-1)(n-2)*...*3*2*1 and using the theorem lim(ks_n) = k*lim(s_n), we can see that the limit approaches infinity as n approaches infinity. Therefore, the statement is true and lim (10^n / n!) = 0.
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Gott_ist_tot
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Homework Statement


Prove using limit theorems that

[tex] lim \frac{10^n}{n!} = 0 [/tex]

Homework Equations


We get to use limit theorems. These include
1 lim(a+b) = lim a + lim b,
2 lim(ab) = lim(a)lim(b),
3 lim(s_n) = [tex] \infty [/tex] iff lim(1/s_n)= 0,
4 lim(ks_n) = k*lim(s_n)
5 if lim(s_n) = [tex] \infty [/tex] and lim(t_n) equals some real number, then lim(s_n*t_n) = [tex] \infty [/tex]

The Attempt at a Solution


I am having difficulty figuring out how to manipulate the factorial to match a theorem. Any advice/hints would be appreciated. Thanks.
 
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  • #2
What does the limit approach? (n---> ?) well expand out n! as n(n-1)(n-2)*...*3*2*1 and see if that helps
 
  • #3
n approached infinity. But what you said did it. Thanks.
 

FAQ: Proving lim 10^n/n!=0 Using Limit Theorems

What is the purpose of proving lim 10^n/n!=0 using limit theorems?

The purpose of proving this limit using limit theorems is to show the behavior of the sequence 10^n/n! as n approaches infinity. It allows us to understand the long-term trend of the sequence and make predictions about its value.

How does limit theorems help in proving the limit of 10^n/n!?

Limit theorems provide a set of rules that can be used to evaluate limits of functions. In this case, we can use the limit theorem for products to simplify the expression 10^n/n! and then apply the limit theorem for quotients to evaluate the limit as n approaches infinity.

Can this limit be proven without using limit theorems?

Yes, there are other methods that can be used to prove this limit, such as the squeeze theorem or the definition of a limit. However, using limit theorems provides a more efficient and straightforward approach.

What is the significance of proving lim 10^n/n!=0?

The limit 10^n/n! is a common limit that appears in many mathematical and scientific applications. Proving that it approaches zero as n approaches infinity allows us to make conclusions about the behavior of related functions and sequences. It also has practical applications in areas such as probability and statistics.

Are there any limitations to using limit theorems in proving this limit?

Limit theorems can only be applied when certain conditions are met, such as the functions being continuous and differentiable. If these conditions are not met, then limit theorems cannot be used to evaluate the limit of 10^n/n!. In such cases, other methods may need to be applied.

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