Convergence and existence of constants

In summary, we discussed the convergence of the sequence $\left(\binom{n}{m}n^{-m}\right)$ and showed that it converges to $\frac{1}{m!}$. From there, we were able to find constants $C_1>0, C_2>0$ and a natural number $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$ for each $n \geq n_0$, demonstrating that the sequence is bounded.
  • #1
evinda
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Hello! (Wave)

Let $m$ be a natural number. I want to check the sequence $\left( \binom{n}{m} n^{-m}\right)$ as for the convergence and I want to show that there exist constants $C_1>0, C_2>0$ (independent of $n$) and a positive integer $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$ for each $n \geq n_0$.

We have that $\binom{n}{m} n^{-m}=\frac{n!}{m!(n-m)!} n^{-m}=\frac{1 \cdots (n-m) \cdot (n-(m-1)) \cdots n}{m!(n-m)!} n^{-m}=\frac{(n-m)! (n-(m-1) \cdots n)}{m!(n-m)!} n^{-m}=\frac{(n-(m-1)) \cdots n}{m!} n^{-m}$.

Is it right so far? If so, how can we find the limit of the last term? (Thinking)
 
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  • #2
evinda said:
Hello! (Wave)

Let $m$ be a natural number. I want to check the sequence $\left( \binom{n}{m} n^{-m}\right)$ as for the convergence and I want to show that there exist constants $C_1>0, C_2>0$ (independent of $n$) and a positive integer $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$ for each $n \geq n_0$.

We have that $\binom{n}{m} n^{-m}=\frac{n!}{m!(n-m)!} n^{-m}=\frac{1 \cdots (n-m) \cdot (n-(m-1)) \cdots n}{m!(n-m)!} n^{-m}=\frac{(n-m)! (n-(m-1) \cdots n)}{m!(n-m)!} n^{-m}=\frac{(n-(m-1)) \cdots n}{m!} n^{-m}$.

Is it right so far? If so, how can we find the limit of the last term? (Thinking)
That is correct so far. You could continue like this (dividing each of the factors in the numerator by one of the $n$s in the denominator):
\[\frac{(n-(m-1)) \cdots n}{m!} n^{-m} = \frac1{m!}\frac{(n-(m-1)) \cdots (n-1)\cdot n}{n^m} = \frac1{m!}\left(1-\frac{m-1}n\right) \cdots \left(1-\frac1n\right).\]
Can you finish it from there?
 
  • #3
Opalg said:
That is correct so far. You could continue like this (dividing each of the factors in the numerator by one of the $n$s in the denominator):
\[\frac{(n-(m-1)) \cdots n}{m!} n^{-m} = \frac1{m!}\frac{(n-(m-1)) \cdots (n-1)\cdot n}{n^m} = \frac1{m!}\left(1-\frac{m-1}n\right) \cdots \left(1-\frac1n\right).\]
Can you finish it from there?

Since $\lim_{n \to +\infty} \frac{x}{n}=0, \forall x \in [1,m-1]$, it follows that $\lim_{n \to +\infty} \binom{n}{m} n^{-m}=\frac{1}{m!}$, right?
 
  • #4
evinda said:
Since $\lim_{n \to +\infty} \frac{x}{n}=0, \forall x \in [1,m-1]$, it follows that $\lim_{n \to +\infty} \binom{n}{m} n^{-m}=\frac{1}{m!}$, right?
Yes, that's right. It's the theorem that says that the limit of a product is the product of the limits. Here, you have a fixed finite number (namely $m-1$) of terms, each of which is converging to $1$, so their product also converges to $1$.
 
  • #5
Opalg said:
Yes, that's right. It's the theorem that says that the limit of a product is the product of the limits. Here, you have a fixed finite number (namely $m-1$) of terms, each of which is converging to $1$, so their product also converges to $1$.

Yes! (Nod)

Since $\lim_{n \to +\infty} \binom{n}{m} n^{-m}=\frac{1}{m!}$, we have that $\forall \epsilon>0$, $\exists n_0 \in \mathbb{N}$ such that $\forall n \geq n_0$,

$$\left| \binom{n}{m} n^{-m}-\frac{1}{m!}\right|< \epsilon \Rightarrow \frac{1}{m!}-\epsilon< \binom{n}{m} n^{-m}<\frac{1}{m!}+\epsilon \Rightarrow \left( \frac{1}{m!}-\epsilon\right) n^m< \binom{n}{m}< \left( \frac{1}{m!}+\epsilon\right)n^m$$

So like that, we have shown that there are constants $C_1>0, C_2>0$ and a natural number $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$ for each $n \geq n_0$, right? (Smile)
 
  • #6
evinda said:
Yes! (Nod)

Since $\lim_{n \to +\infty} \binom{n}{m} n^{-m}=\frac{1}{m!}$, we have that $\forall \epsilon>0$, $\exists n_0 \in \mathbb{N}$ such that $\forall n \geq n_0$,

$$\left| \binom{n}{m} n^{-m}-\frac{1}{m!}\right|< \epsilon \Rightarrow \frac{1}{m!}-\epsilon< \binom{n}{m} n^{-m}<\frac{1}{m!}+\epsilon \Rightarrow \left( \frac{1}{m!}-\epsilon\right) n^m< \binom{n}{m}< \left( \frac{1}{m!}+\epsilon\right)n^m$$

So like that, we have shown that there are constants $C_1>0, C_2>0$ and a natural number $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$ for each $n \geq n_0$, right? (Smile)
Good! :)

The only thing you have to be careful about is to choose $\epsilon < \frac1{m!}$ (otherwise $C_1$ would not be positive).
 
  • #7
Opalg said:
Good! :)

The only thing you have to be careful about is to choose $\epsilon < \frac1{m!}$ (otherwise $C_1$ would not be positive).

I see... Thanks a lot... (Smile)
 

FAQ: Convergence and existence of constants

1. What is convergence and existence of constants?

Convergence and existence of constants is a mathematical concept that deals with the behavior of sequences and series of numbers. It focuses on whether these sequences and series approach a specific value, known as a constant, and whether this constant actually exists.

2. What does it mean for a sequence or series to converge?

A sequence or series converges if its terms get closer and closer to a specific value, known as the limit, as the number of terms increases. In other words, the sequence or series approaches a fixed value and stays close to it.

3. How is convergence related to the existence of constants?

The existence of a constant is dependent on whether the sequence or series converges. If the sequence or series converges, then the constant exists. If the sequence or series does not converge, then the constant does not exist.

4. What is the difference between absolute convergence and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which its terms are added. On the other hand, conditional convergence refers to a series that only converges when its terms are added in a specific order.

5. Why is the convergence and existence of constants important?

Convergence and existence of constants are important in mathematics, as they provide a way to determine the behavior of sequences and series and whether a constant actually exists. This concept is also used in various applications in physics, engineering, and other fields to analyze and solve problems.

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