Question

okay guys I have a calc 2 question:

how should I solve this ratio test?

Σ k^20 (x)^k /(2k+1)!

Answer 1

Permutation or combination ?

Answer 2

is k^20 (x)^k multiplying?

Answer 3

yes

Answer 4

this is a series but they want to find out the interval and radius of convergence which I know how to figure that out but my main problem is to do the ratio test properly.

Answer 5

ok for finding the radius of convergence the ratio test should be <1
Lim \[\lim_{k \rightarrow \infty}(ak+1)/ak \]
ak =k^20 (x)^k /(2k+1)!

Answer 6

now to get (ak+1) substitute k by k+1

Answer 7

okay yeah you get something like \[k^20+1 x^k+1 / (2k+2)\]

Answer 8

x^k+1 i meant

Answer 9

hmmm no.. it would be something like this
[(k+1)^20 * (x)^k+1] / (2(k+1) + 1)!
[(k+1)^20 * (x)^k+1] / (2k+ 3)!
do u get it?

Answer 10

this is ur ak+1 @rayjan

Answer 11

let me see one min budd

Answer 12

sure and remember its all this will be written inside mod and its all less then 1 bcos it converges..

Answer 13

(2k +3 )! can also be written as (2k+3)(2k+2)(2k+1)!

Answer 14

just a quick question budd about (2k+3)! well the way they explained it to us is that you just add 1 to 2k+1 which would be 2k+2! so I am a little confused there and also how would I cancel my terms I mean what goes away here comparing to original equation?

Answer 15

no .. it adds 1 into k.. means every k would become k+1

Answer 16

oh got it so now we have (k+1)^2*x^k+1/(2k+3)! * the original equation so can you please help me to see which terms would cancel?

Answer 17

yes x^ k+1 can also be written as x^k * x^1..? do u get this?

Answer 18

ahm no honestly

Answer 19

i have just break it tell me if u have
x^3*x^2 what wil u get?

Answer 20

x^5

Answer 21

yes and how u get that? u did x^ (3+2) right?

Answer 22

right

Answer 23

so same way.. x^(k+1) can be written x^k * X^1 right?

Answer 24

yes

Answer 25

basically when u have same term and its multiplying u just add the powers.

Answer 26

ok when u apply ratio rule.. x^k will be cancell.. please keep doing on page so u can know what are u doing

Answer 27

okay one min.

Answer 28

okay I got it in this form now:\[k^20*x^k/(2k+1)! * (2k+3)!/(k+1)^20*x^k+1\]