Square root n limit ( sum question )

In summary, the conversation discusses solving for the limit of a sequence and the approach to solving a specific sum that is needed for the limit calculation. The sum is shown to be equal to (n! - 1) and the conversation ends with a grateful response.
  • #1
Vali
48
0
Hi!

$$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$
$$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$
I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty } \sqrt[n]{\frac{n!}{n^{n}}}$ which I know how to solve.
So, how to approach the sum such that the result to be $(n! - 1)$ ?
 
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  • #2
Hello again, Vali. (Wave)

Note $(k-1)\cdot (k-1)! = k(k-1)! - (k-1)! = k! - (k-1)!$, and thus the sum $\sum\limits_{k = 2}^n (k-1)\cdot (k-1)! = \sum\limits_{k = 2}^n [k! - (k-1)!]$ telescopes to $n! - 1$.
 
  • #3
Thank you! :)
 

Related to Square root n limit ( sum question )

1. What is the definition of a square root limit?

A square root limit is a mathematical concept that involves finding the value that a function approaches as the input (n) gets closer and closer to a specified number. In other words, it is the value that the function "squares" to in order to get the input value.

2. How do you solve a square root limit?

To solve a square root limit, you can use the limit laws and algebraic manipulation to simplify the expression. Then, you can use the properties of limits to evaluate the limit. Alternatively, you can use L'Hopital's rule or Taylor series to find the limit.

3. What is the difference between a square root limit and a regular limit?

A square root limit specifically deals with finding the limit of a function that contains a square root. In a regular limit, the function can contain any type of mathematical operation.

4. Can a square root limit have a negative value?

Yes, a square root limit can have a negative value. This can happen when the function approaches a negative number as the input gets closer and closer to the specified number.

5. How is a square root limit used in real life?

Square root limits are used in various fields of science and engineering to model and analyze natural phenomena. For example, in physics, square root limits are used to calculate the velocity of a falling object or the rate of change of a chemical reaction. In finance, square root limits are used to calculate the expected return on an investment.

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