Question

does the series converge or diverge?

(2n)!/(n+3)! n=1 to infinity

Answer 1

did you try ratio test ?

Answer 2

let me do that now:)

Answer 3

An=(2n)!/(n+3)!
An+1=(2n+1)!/((n+3)+1)!

??

Answer 4

correct

Answer 5

((2n+1)!/((n+3)+1)!)/((2n)!/(n+3)!)

Answer 6

wait,
2n -------> 2 (n+1) = 2n+2

Answer 7

oh okay! i thought something would be wrong haha

Answer 8

and use the fact that
(x+1)! = (x+1)x!

Answer 9

(2n+2)! = (2n+2)(2n+1)2n!
(n+4)! = (n+4)(n+3)!

Answer 10

((2n+2)!/(2n)!)*((n+3)!/(n+3)+1)!) how do i simplify this?

Answer 11

wait what?

Answer 12

did you know this ?
(x+1)! = (x+1)x!
?

Answer 13

nope:/

Answer 14

how will you get say umm, 4! =.. ?

Answer 15

4*3*2*1

Answer 16

3! = 3*2*1
so,
4! = 4*3!
this is true in general

Answer 17

n! = n (n-1)!

Answer 18

makes sense:)

Answer 19

so, (2n+2)! = (2n+2) (2n+1)!

Answer 20

and (2n+1)! = (2n+1)2n!

Answer 21

so you got this ?
(2n+2)! = (2n+2)(2n+1)2n!
(n+4)! = (n+4)(n+3)!

Answer 22

i think so.....also, is (n+3)+1) simplified to just n+4?

Answer 23

yes, thats just
(n+1)+3 = n+4

Answer 24

ok just checking

Answer 25

do i have to use that rule for (2n)! and (n+3)! too?

Answer 26

those will get cancelled , right ?

Answer 27

yes...i see it now lol

Answer 28

(2n+2)(2n+1)/n(n+4)

Answer 29

how does "n" come in the denominator ?

Answer 30

typo sorry!

Answer 31

no problem
can you evaluate
lim n->infinity (2n+2)(2n+1)/(n+4)

Answer 32

yeah hold on

Answer 33

*holding tight*

Answer 34

haha....had to get my drink! isnt it infinity bc as n in the denominator gets bigger the entire den. gets bigger? approaching infinity?

Answer 35

so it diverges?

Answer 36

absolutely correct!

Answer 37

yes

Answer 38

yay!!!! omgsh thank you so much! i'm so going to get this problem right if its on my exam!

Answer 39

glad to hear!
ok, just an algebraic way of solving the limit
multiply numerator and denominator by 1/n ^2
as n-> infinity , 1/n -> 0, 1/n^2-> 0
so you will get a 0 in the denominator , implies lim = infinity

Answer 40

okay!!
now i have a question....how do u know when to n(n-1)!? bc we didnt use it on 2n! and (n+3)!

Answer 41

we had (2n+2)! in numerator and 2n! in denominator, right ?

its like having 5! in numerator and 3! in denominator,
so the only thing we can use is n! = n(n-1)! to simplify

Answer 42

okay so u just look at it and think well if i simplify this by using the formula it will cancel that out!

Answer 43

correct ...
and when you do few more problems like this, you'll get what to do automatically in other similar problems :)

Answer 44

can u work on another with me? i already worked it out but idk if its correct

Answer 45

im studying for my exam this wednesday and there isnt an answer key to my practice problems so idk if im getting any of these correct or not:(