Question

series expansion question:
find the power series expansion for the function
x/(2-x) and find the radius of convergence.

Answer 1

I know the power series formula is 1/1-x and my study guide says to get there factor out the 1/2 in x/(2-x)....which is ok but the next step in my guide just disappears the x on top like this:
x/2−x=(1/2)∗1/(1−(x/2))


howwhywhat happened to the x in the numerator?

Answer 2

i would do long division to put this into a power series

Answer 3

x/2 +x^2/4+x^3/8 ....
------------------
2-x ) x
(x-x^2/2)
-----------
x^2/2
(x^2/2 -x^3/4)
---------------
x^3/4

Answer 4

as such we notice a pattern

\[\sum_{n=1}^{inf}\frac{x^{n}}{2^n}\]

Answer 5

heh, YOU notice a pattern =-)

Answer 6

http://www.wolframalpha.com/input/?i=sum+x%5En%2F2%5En+%2C+n%3D1+to+inf

urgh .... i got another negative lol

Answer 7

not again...lol

Answer 8

we have singular point is 2
we need center point?

Answer 9

using this summation we can find the rest of it

Answer 10

\[\lim_{n\to\ inf}\frac{-x^n}{2^n}\frac{2^{n-1}}{-x^{n-1}}\]

Answer 11

the negatives cancel and it looks like all our ns disapear to make this a 1

Answer 12

\[|\frac{x}{2}|<1\]
\[|x|<2\]

Answer 13

as such, our radius is 2 and our interval of convergence is -2 to 2

Answer 14

If you think of the x^n/2^n as (x/2)^n
absolute value of x/2 converges when less than 1 and its less than 1 when x < 2

as amistre64 says

Answer 15

lol, one of these days i mmight even figure out what im doing :)

Answer 16

we have singular point is 2
if power series expansion at 0
->radius is 2

Answer 17

sorry chanh, i got no way to verify your method :/

Answer 18

at least not in generality

Answer 19

and the disappearing n's (and the corresponding -x) explains why the x's disappeared in my study key they just skipped like one billion steps

Answer 20

skipping steps saves on ink

Answer 21

I'm thinking of just drawing a black box on my test labelled magic - and putting the answer next to it.

Answer 22

the thunder was a nice touch, if you can recreate that effect during the test, your sure to get an A lol

Answer 23

I'm study in complex variables with applications book

Answer 24

hmmm I dont know how that lines up with Early Transcendentals