- #1
MidgetDwarf
- 1,523
- 672
So i am given (1+x)/(1-x)^2 and I have to put it into a power series. I know that 1/(1-x)= 1+x+x^2+x^3+...=∑x^n from 0 to infinity. I am having problems factoring series.
I differentiate 1/(1-x).
I get, 1/(1-x)^2= 1+2x+3x^2+...= ∑nX^(n-1) the sum from 1 to infinity.
rewriting this equation.
1/(1-x)^2=∑(n+1)X^n. The sum from 0 to infinity.
multiply both sides by (1+x)
I get (x+1)/(1-x)^2 = (1+x)∑(n+1)X^(n) , from 0 to infinity.
Then I distribute 1+x.
∑(n+1)X^(n) + ∑(n+1)X^(n+1) both sums have an index of 0.
My problem is that I have having trouble factoring using summation notation and I am not sure how my book factored these 2 sums into 1 to get the answer
∑(2n+1)X^n with the index of 0.
When I factored my previous work I got. ∑(n+1)(X^(n)[1+X]
which I cannot seem to to put into the form the book has it.
I differentiate 1/(1-x).
I get, 1/(1-x)^2= 1+2x+3x^2+...= ∑nX^(n-1) the sum from 1 to infinity.
rewriting this equation.
1/(1-x)^2=∑(n+1)X^n. The sum from 0 to infinity.
multiply both sides by (1+x)
I get (x+1)/(1-x)^2 = (1+x)∑(n+1)X^(n) , from 0 to infinity.
Then I distribute 1+x.
∑(n+1)X^(n) + ∑(n+1)X^(n+1) both sums have an index of 0.
My problem is that I have having trouble factoring using summation notation and I am not sure how my book factored these 2 sums into 1 to get the answer
∑(2n+1)X^n with the index of 0.
When I factored my previous work I got. ∑(n+1)(X^(n)[1+X]
which I cannot seem to to put into the form the book has it.