Evaluate `lim_(x to 1) sum_(k=1)^(100) x^(k) - 100)/(x-1).`

Answer

5050

Explanation

`underset(x to 1)("lim")([sum_(k=1)^(100)x^(k)] - 100)/((x-1))`
`=underset(xto1)lim((x+x^(2)+x^(3)+...+x^(100))-100)/((x-1))`
`=underset(xto1)lim((x-1)+(x^(2)-1)+(x^(3)-1)+...+(x^(100)-1))/((x-1))`
`=underset(xto1)lim{((x-1)/(x-1))+((x^(2)-1)/(x-1))+((x^(3)-1)/(x-1))+....+((x^(100)-1)/(x-1))}`
`=underset(xto1)lim{((x-1)/(x-1))+underset(xto1)lim((x^(2)-1)/(x-1))+underset(xto1)lim((x^(3)-1)/(x-1))+...`
`underset(xto1)lim((x^(100)-1)/(x-1))=1+2+3+...+100`
`=(100xx101)/(2)=5050`