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