Question

1+1/10+1/100+1/1000+------n

Answer 1

since x = 1/10 < 1,

1 + x + x^2 + x^3 + x^4 +.... = Σ(k = 0 to ∞) x^k = 1/(1 - x)

= 1/(1 - 1/10) = 1/(10/10 - 1/10) = 1/(9/10) = 10/9

of course, writing the series as the infinite repeating decimal one has:

r = 1.111111........
10r = 11.111111........

10r - r = 10
9r = 10
r = 10/9