Question

Start with the geometric power series LaTeX: 1+x+x^2+x^3+x^4+\cdots Substitute LaTeX: -x^3in place of LaTeX: x, and then take a derivative term by term. Express this new series by writing out at least the first five terms, and also express it using summation notation.

Answer 1

`-3x^2+6x^5-9x^8+12x^(11)+...=\sum_(n=1)^(\infty )\left ( -1 \right )^n\,\,3n\,\,x^(3n-1)`

Explanation:

Given :
`1+x+x^2+x^3+x^4+\cdots`

On putting
`-x^3` in place of x , we get
`1+\left ( -x^3 \right )+\left ( -x^3 \right )^2+\left ( -x^3 \right )^3+\left ( -x^3 \right )^4+...`

On simplifying , we get
`1-x^3+x^6-x^9+x^(12)+...`

On differentiating , we get
`\left ( 1-x^3+x^6-x^9+x^(12)+... \right )'=-3x^2+6x^5-9x^8+12x^(11)+...`

Here ,

`-3x^2=\left ( -1 \right )^1\,3(1)x^(3(1)-1)\\6x^5=\left ( -1 \right )^2\,\,3(2)\,\,x^(3(2)-1)\\-9x^8=\left ( -1 \right )^3\,\,3(3)\,\,x^(3(3)-1)\\12x^(11)=\left ( -1 \right )^4\,\,3(4)\,\,x^(3(4)-1)`

Now , we need to express it using summation notation.

`\left ( 1-x^3+x^6-x^9+x^(12)+... \right )'=-3x^2+6x^5-9x^8+12x^(11)+...=\sum_(n=1)^(\infty )\left ( -1 \right )^n\,\,3n\,\,x^(3n-1)`