Question

What is the sum of the series `1+(1+x)+(1+x+x^2)+(1+x+x^2+x^3)+..."to n terms"`?

Answer 1

`S_n=n/(1-x)-(x(1-x^n))/(1-x)^2`

Explanation:

Let `S_n=1+(1+x)+(1+x+x^2)+(1+x+x^2+x^3)+.....` to `n` terms.

Hence, `(1-x)S_n=(1-x)+(1-x)(1+x)+(1-x)(1+x+x^2)+(1-x)(1+x+x^2+x^3)+....` to `n` terms

or `(1-x)S_n=(1-x)+(1-x^2)+(1-x^3)+(1-x^4)+......` to `n` terms

`=n-(x+x^2+x^3+x^4+.......)` to `n` terms

`=n-(x(1-x^n))/(1-x)`

Hence `S_n=n/(1-x)-(x(1-x^n))/(1-x)^2`

Note that if `x>1` this becomes

`S_n=(x(x^n-1))/(x-1)^2-n/(x-1)`