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

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Answer 1 · 2017-07-19T09:07:56

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

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)#