Question

If x<1 then find the sum to n terms of the senes (1+x)+(1+x+x^(2))+(1+x+x^(2)+x^(3))+...cdots​

Answer 1

Answer:

n/(1-x) - (x^2(1-x^n))/(1-x)^2

Step-by-step explanation:

Let S = (1+x)+(1+x+x^(2))+(1+x+x^(2)+x^(3))+...

(1-x)S =(1-x)(1+x)+(1-x)(1+x+x^(2))+(1-x)(1+x+x^(2)+x^(3))+...

(1-x)S= 1- x^2 + 1-x^3 + 1- x^4+...+ 1-x^n+1

(1-x)S=n-(x^2+x^3+x^4+...+x^n+1)

(1-x)S= n-(x^2(1-x^n))/(1-x)

S=n/(1-x) - (x^2(1-x^n))/(1-x)^2