Question

expand f(x)=[x+sqr(x)]/[(1-x)^3] as a power series and use it to find the sum of sigma n^2/2^n

Answer 1

I think I would try to get the bottom rationalized

Answer 2

nah, expanded; since ^3 aint a radical :)

Answer 3

then divide it out

Answer 4

1 - x - x^3 i beleive is the bottom expanded

Answer 5

er, +x

Answer 6

x - x^2
--------------------
1 + x - x^3 ) x + sqrt(x)
(x+x^2-x^4)
-------------
-x^2+x^4+sqrt(x)
(-x^2-x^3+x^5)
-----------------
x^3+x^4 +x^5 +sqrt(x)

hmmm, should we let x^1/2 come before x ? or continue as planned?

Answer 7

hi this is a power series question

Answer 8

if anyting we could split the numberator

Answer 9

yes, and this is one way to build a power series .... or do you have something more specific in mind?

Answer 10

yah...derivative of 1/1-x gives 1/(1-x)2 and again differentiating gives 1/(1-x)^3
like this way we can solve

Answer 11

then determine the derivatives of f(x) till you see a pattern emerget hat can notated in summation

Answer 12

can u answer my second part of the question

Answer 13

let me consider it .... see if i can make sense of what is being asked :)

Answer 14

ok

Answer 15

when I split up the numerator into x and sqrt(x) I get similar expansions:

\[\frac{x}{(1-x)^3}=x+3x^2+6x^3+10x^3+15x^4+21x^5...\]
\[\frac{x^{1/2}}{(1-x)^3}=x^{1/2}+3x^{3/2}+6x^{5/2}+10x^{7/2}+15x^{9/2}+21x^{11/2}...\]

Answer 16

but I aint sure how to use it to find the second part