Question

Find the radius of convergence and interval of convergence of the series \[\sum_{n=1}^{∞}\] 2^n(x-2)^n / (n+2)!

Answer 1

i am getting that it converges no matter what.

Answer 2

by ratio test. but i could be wrong so let me try again

Answer 3

can u show me the steps u did?

Answer 4

\[\frac{a_{n+1}}{a_n}=\frac{2(x-2)}{n+3}\]

Answer 5

oh n+3

Answer 6

i computed the ratio and that is what i got.
\[a_{n+1}=\frac{2^{n+1}(x-2)^{n+1}}{(n+3)!}\]

Answer 7

ok thanks

Answer 8

so
\[\frac{a_{n+1}}{a_n}=\frac{2^{n+1}(x-2)^{n+1}(n+2)!}{(n+3)!2^n(x-2)^n}\]

Answer 9

after canceling my fool head off i got
\[\frac{a_{n+1}}{a_n}=\frac{2(x-2)}{n+3}\]

Answer 10

whose limit as n goes to infinity is 0 regardless of x. you should check my algebra though!

Answer 11

yah its between 1/2 < x < 7/2 when it converges

Answer 12

on the other hand n! grows way faster than 2^n so it is not surprising that this would converge

Answer 13

really? then i made a mistake. i wonder what it was?

Answer 14

what did you do?

Answer 15

wait my bad its 0<x<3/2

Answer 16

what did you do with this
\[\frac{a_{n+1}}{a_n}=\frac{2(x-2)}{n+3}\]?

Answer 17

yah that but i have to find the interval

Answer 18

well it has been a while but as i recall you just have to make sure that for fixed x the the limit as n goes to infinity of this thing has to be less than 1 in absolute value. for a fixed x this limit is zero. so unless i forgot something important this converges for all x

Answer 19

you will notice that there is no "n" in the numerator. so i think it converges for all x

Answer 20

almost positive it does in fact.

Answer 21

i certainly converges for x = 10 because i just tried it
look here
http://www.wolframalpha.com/input/?i=sum+2^n%2810-2%29^n+%2F+%28n%2B2%29!

Answer 22

ohhh ok

Answer 23

usually you get something like


\[\frac{nx}{2n}\] and then you have to make sure that\[|\frac{nx}{2n}|\leq1\] so
\[-2 \leq x \leq 2\] is your radius. but in your example the limit is zero no matter what. just thought i would mention it

Answer 24

thanks