Question

can anyone help out with power series?

Answer 1

Yes

Answer 2

so waht is your question?

Answer 3

Find the radius of convergence and the interval of convergence of the power series.
the sum from n=0 to infinity of 2^n x^n / n!

Answer 4

so i used ratio test and found it converges absolutely, but im not sure how to find the Radius on convergence or interval

Answer 5

Do you mind giving me the last few lines of your ratio test?

Answer 6

well it boiled down to lim=|2x/n+1|

Answer 7

COOL i got that too

Answer 8

so limn->inifinity, 2x/(n+1) -> 0

Answer 9

yupp, exactly

Answer 10

that mean the interval is (-infin, infin) and radius of convergence is infinity

Answer 11

that means*

Answer 12

ok, just so i understand though what is the R value

Answer 13

oh is it infinity

Answer 14

yes

Answer 15

so how was i able to decipher that the R value is infinity just from knowing the series converged absolutely?

Answer 16

i guess im asking why does geting L=0 from the ratio test help me

Answer 17

u can interpret it this way lim n->infin |x|(2/(n+1))<1 , then |x|<1/ 0 |x|<infinity=R

Answer 18

X is a constant. So no mater what X is, the limit will go to 0, and the ratio test states that if the limit is <1, it converges. So X can be any real numbers

Answer 19

ah ok, so becasue this series converges no matter what x-value is plugged in: my x value can range from -infinity to +infinity bc it will always converge. and then the radius for that interval is infinity also

Answer 20

YEP that is it!!

Answer 21

thankyou very much i appreciate it. and just for learning purposes, do i know this series is centered at 0?

Answer 22

yes

Answer 23

and that is bc the x-a isnt there?

Answer 24

Becasue it is the form of c0+c1(x)+c2(x)^2 +....cn(x)^n

Answer 25

yep!!

Answer 26

so its in the c0 form

Answer 27

i think i get this

Answer 28

yea cn(x)^n then it is centered at 0, if it is cn(x-1)^n then it would be centered at x=1

Answer 29

and in my equation here the 2^2/n! is the Cn part

Answer 30

2^n/n!

Answer 31

u got it?

Answer 32

right right ok im dumb

Answer 33

LOL no i know that is you meant.

Answer 34

2^2 is just a typo~~

Answer 35

yup my fault

Answer 36

well i really do appreciate this. It may not seem it but you just helped someone who will most likely be getting early admission to UB med school ha ha Ive been getting pounded with organic and inorganic so this has been on the back burner

Answer 37

LOL my pleasure and honor :P

Answer 38

keep up the good work you are really patient, and a good listener. Guys like you are doing it for the love of the game if you will.

Answer 39

LOL alright if u have any further questions, just post here :) i am gonna go now
Bye :)

Answer 40

oh can you find the radius of convergence for this power series : sum n = 0 to n =infinity ((x+3)^n)/(e^3n!)

Answer 41

welp i think the ratio test gives you the limit to approach zero again, so i guess its infinity again?

Answer 42

Thanks lol i was arguing with my friend about the convergence of this series.

Answer 43

yeah like at the end of my ratio test i got x+3 ontop

Answer 44

but the numerator is like e^3n+1 time e^3n+2 times e^3m+3 so thats infinity

Answer 45

i just meant denominator* sorry so basicall that makes the denom infinity and the numerator x+3

Answer 46

yea haha Thanks!~