Question

Limits of series. I am going to type the equation using equation editor.

Answer 1

I am trying to find the following limit.

\[\lim_{n \rightarrow \inf} \frac{ 2(n!) }{ (-6)^n }\]

Answer 2

Factorial will grow much much bigger than any exponential growth (after some point).

Answer 3

Is it because the factorial is multiplied by 2?

Answer 4

I was under the assumption that the (-6)^n would grow faster than the factorial.

Answer 5

two does not do anything at all.... whether it has a two or not factorial is much bigger.

Wrong assumption! The opposite is true

Answer 6

Then I guess the limit diverges to both negative and positive infinity.

Answer 7

Yes:)
positive and negative infinity, because it alternates.

Answer 8

Alright, Thanks. I guess my understand of factorials are a bit off.

Answer 9

We can play around and maybe apply Stirling's approximation:
(Note, that the factorial is greater than its stirlings approximation)

\(\large\color{black}{ \displaystyle n!\approx\frac{\sqrt{2\pi n}{~}n^n}{e^n} }\)

\(\large\color{black}{ \displaystyle \frac{n!}{6^n}\approx\frac{\sqrt{2\pi n}{~}n^n}{6^ne^n} }\)

\(\large\color{black}{ \displaystyle \frac{n!}{6^n}\approx\sqrt{2\pi n}\frac{{~}n^n}{(6e)^n} }\)

Answer 10

thx i learned something new just by watching lol

Answer 11

You would then agree that the numerator is greater, right?

Answer 12

I do agree that the numerator is greater.

Answer 13

Good:)

And I understand that you were applying the Diverghence test to
\(\large\color{black}{ \displaystyle \sum_{ n=1 }^{ \infty } ~ \frac{2(n!)}{(-6)^n}}\)

Answer 14

(not necessarily n=1, but that series)

Answer 15

yep

Answer 16

Ok, so if it was the other way around, then you would apply:

\(\large\color{black}{ \displaystyle \sum_{ n=0 }^{ \infty } ~ \frac{x^n}{n!}=e^x}\)

So for example:

\(\large\color{black}{ \displaystyle \sum_{ n=0 }^{ \infty } ~ \frac{(-6)^n}{n!}=e^{-6}= 1/e^6}\)

Answer 17

Anyway, good luck....

Answer 18

Thanks for the help. It really made since. I am now making forward progress on my assignments.

Answer 19

Good to hear that:)

Answer 20

be well