Hello, I was wondering if anyone can help me with this problem.
Find a closed form for
/sum_{k=0}^n x^k where x does not equal to zero.
Then prove that it is valid for all integers n>= 1.
This is the geometric sum:
1+x+x^2+...+x^n
Let S=1+x+...+x^n
Then,
xS=x+x^2+...+x^{n+1}
Thus,
S-xS=(1+x+...+x^n)-(x+x^2+...+x^{n+1})=1-x^{n+1}
S(1-x)=1-x^{n+1}
Now if x is not equal to 1 then,
S=(1-x^{n+1})/(1-x)
And if x is equal to 1 then,
1+x+x^2+...+x^n=1+1+...+1=n
Thus,
sum[k=0,n]x^k = (1-x^{n+1})/(1-x) if x not = 1, n if x =1.