Solve:

$\displaystyle\sum _{ n=1 }^{ \infty }{ \frac { 9 }{ { 10 }^{ n } } }$

The answer is 1.

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Well it's just an Infinite Geometric Progression

a_1 = 9/10

common ratio of 1/10

S= a_1 / 1-r

S= (9/10) / [1-(1/10)]

S= (9/10) / [9/10]

S=1