Question

find \[\sum_{i=0}^{10} 0.9^i\]

Answer 1

6.861894039

Answer 2

how did you do it ?

Answer 3

\[\huge Very *long\]
S[i=0,10,0.9^(i)]

S[i=0,10,0.9^(i)]=1+0.9^(1)+0.9^(2)+0.9^(3)+0.9^(4)+0.9^(5)+0.9^(6)+0.9^(7)+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.9^(3)+0.9^(4)+0.9^(5)+0.9^(6)+0.9^(7)+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.9^(4)+0.9^(5)+0.9^(6)+0.9^(7)+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.9^(5)+0.9^(6)+0.9^(7)+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.5905+0.9^(6)+0.9^(7)+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.5905+0.5314+0.9^(7)+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.5905+0.5314+0.4783+0.9^(8)+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.5905+0.5314+0.4783+0.4305+0.9^(9)+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.5905+0.5314+0.4783+0.4305+0.3874+0.9^(10)

S[i=0,10,0.9^(i)]=1+0.9+0.81+0.729+0.6561+0.5905+0.5314+0.4783+0.4305+0.3874+0.3487

S[i=0,10,0.9^(i)]=6.8619

Answer 4

\[\sum\limits_{i=0}^{10}0.9^i=\frac{1-0.9^{11}}{1-0.9}=\frac{1-0.9^{11}}{0.1}=10(1-0.9^{11})\approx 6.862\]