what is the sum of the series ?

[NJ_84]

<sup>what is the sum of series:
((n-3)/n)^n

im stuck after : (1-3/n)^n
which test do i use?</sup>

 

[NJ_84]

is it just equals to e ?

 

[soroban]

Hello, NJ_84!

Recall the definition: .\(\displaystyle \displaystyle \lim_{x\to\infty}\left(1 + \frac{1}{x}\right)^x \;=\;e\)

\(\displaystyle \displaystyle \lim_{n\to\infty} \left(\frac{n-3}{n}\right)^n\)

\(\displaystyle \left(\frac{n\:-\:3}{n}\right)^n \;=\;\left(1 + \frac{3}{n}\right)^n \;=\;\left[\left(1 + \frac{3}{n}\right)^{\frac{n}{3}}\right]^3 \;=\; \left[\left(1 + \frac{1}{\frac{n}{3}}\right)^{\frac{n}{3}}\right]^3 \)

\(\displaystyle \displaystyle\text{Therefore: }\:\lim_{\frac{n}{3}\to\infty} \left[\left(1 + \frac{1}{\frac{n}{3}}\right)^{\frac{n}{3}}\right]^3 \;=\; \left[\lim_{\frac{n}{3}\to\infty} \left(1 + \frac{1}{\frac{n}{3}}\right)^{\frac{n}{3}}\right]^3 \;=\;e^3\)

 

[NJ_84]

thank you soroban !!!!!!!

 

[HallsofIvy]

So the limit of the sequence is e. What does that tell you about the sum of the series?

 

[lookagain]

<sup>what is the sum of series:
((n-3)/n)^n

im stuck after : (1-3/n)^n
which test do i use?</sup>

NJ_84,

you didn't type a series.

You typed a general term of the sequence.

You didn't type a beginning value for n, nor did you type an end value for n.


\(\displaystyle Look \ \ at \ \ \ \displaystyle\sum_{n = 1}^{\infty} \ \bigg(\dfrac{n - 3}{n}\bigg)^n\)


Or, if you had typed something along the lines of the following,
then that would have indicated a series:


Summation (n = 1 to oo) ((n - 3)/n)^n