Question

Consider the series 2n(n!) / n^n Evaluate the following limit

Answer 1

\[\sum_{\infty}^{n=1} \frac{2^n( n!) }{n^n }\]

Answer 2

Well i was able to simplify it to \[ \frac{ 2n^n }{ (n+1)^n }\] but i don't know how to proceed

Answer 3

usin the ratio test

Answer 4

are you tryig to apply ratio test ?

Answer 5

yes

Answer 6

\(\large
\lim\limits_{n\to\infty} \frac{ 2n^n }{ (n+1)^n }\\~\\
=2\lim\limits_{n\to\infty} \left(\frac{ n}{ n+1 }\right)^n\\~\\

=2\lim\limits_{n\to\infty} \left(\frac{ n+1-1}{ n+1 }\right)^n\\~\\
=2\lim\limits_{n\to\infty} \left(1+\frac{ -1}{ n+1 }\right)^n\\~\\
=2*e^{-1}\\~\\
\lt 1
\)

so the series converges by ratio test

Answer 7

thanks, but could you explain where the e came from?

Answer 8

you must be knowing the limit definition of the function \(e^x\) :
\[\lim\limits_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x\]

Answer 9

im pretty sure it bothers you that we have \(n+1\) in the bottom and not \(n\)

Answer 10

but first, just so you get used to that limit definition of \(e^x\),the limit converges for all real values of \(x\). so plugging in \(x=-1\) gives you

\[\lim\limits_{n\to\infty} \left(1+\frac{\color{red}{-1}}{n}\right)^n = e^{\color{red}{-1}} \]

Answer 11

Okay, but the two ns aren't the same since one is n+1 is there a rule that lets us turn it to n?

Answer 12

Here is one way to evaluate \(\lim\limits_{n\to\infty} \left(1+\frac{ -1}{ n+1 }\right)^n \) step by step:

\(\lim\limits_{n\to\infty} \left(1+\frac{ -1}{ n+1 }\right)^n \\~\\
=\lim\limits_{n\to\infty} \left(1+\frac{ -1}{ n+1 }\right)^{n+1} \left(1+\frac{ -1}{ n+1 }\right)^{-1} \\~\\
=\lim\limits_{t\to\infty} \left(1+\frac{ -1}{t }\right)^{t} * \lim\limits_{n\to\infty}\left(1+\frac{ -1}{ n+1 }\right)^{-1} \\~\\
=e^{-1}*\left( \lim\limits_{n\to\infty}1+\frac{ -1}{ n+1 }\right)^{-1} \\~\\
=e^{-1}*(1+0)^{-1}\\~\\
=e^{-1}
\)

Answer 13

there is no general rule that tells us to replace "n+1" by "n", we need to be careful when we do such things... intuition doesn't work in our favor always...

Answer 14

Holy cow! I didn't even know that it was possible to manipulate problems like this. Thank you so much. I can follow it and it makes sense now. I'm blown away by how good you are. Thanks a million.

Answer 15

np, these convergence tests give you very good practice with limits too.... have fun :)