# Thread: sum of infinite series of functions

1. ## sum of infinite series of functions

Find the function, f(x), which the infinite series
1-(x^3)+(x^6)-(x^9)+(x^12)-(x^15)+....... converges to.
The hint is "What is 1-x+(x^2)-(x^3)+(x^4)-(x^5)+....."

I know that 1-x+(x^2)-(x^3)+(x^4)-(x^5)+.... sums to 1/(1+x) for lxl<1 because it is the derivative of the taylor series for ln(1+x) but all I can think of to use that hint is that f(x)=(1/(1+x))+x-(x^2)-(x^4)+(x^5)+(x^7)-(x^8)-(x^10)+(x^11)+.... which doesn't help me much (as far as I can tell).

2. Originally Posted by 234578
Find the function, f(x), which the infinite series
1-(x^3)+(x^6)-(x^9)+(x^12)-(x^15)+....... converges to.
The hint is "What is 1-x+(x^2)-(x^3)+(x^4)-(x^5)+....."

I know that 1-x+(x^2)-(x^3)+(x^4)-(x^5)+.... sums to 1/(1+x) for lxl<1 because it is the derivative of the taylor series for ln(1+x) but all I can think of to use that hint is that f(x)=(1/(1+x))+x-(x^2)-(x^4)+(x^5)+(x^7)-(x^8)-(x^10)+(x^11)+.... which doesn't help me much (as far as I can tell).
Since $\displaystyle 1-x+x^2-x^3+x^4-x^5+\ldots=\frac{1}{1+x}$, take note that

$\displaystyle 1-x^3+x^6-x^9+x^{12}-x^{15}+\ldots=1-(x^3)+(x^3)^2-(x^3)^3+(x^3)^4-(x^3)^5+\ldots=\ldots$

I leave it for you to finish this problem.

3. Originally Posted by Chris L T521
Since $\displaystyle 1-x+x^2-x^3+x^4-x^5+\ldots=\frac{1}{1+x}$, take note that

$\displaystyle 1-x^3+x^6-x^9+x^{12}-x^{15}+\ldots=1-(x^3)+(x^3)^2-(x^3)^3+(x^3)^4-(x^3)^5+\ldots=\ldots$

I leave it for you to finish this problem.

ohhhhh!! thank you very much. I think I need more sleep, I should have seen that.