1. ## Infinite geometric series

Use the sum of an infinite geometric series to prove that o.101010...=10/99. Notice how this decimal expression alternates between 0 and 1 indefinitely, so essentially we do not know whether it terminates on 0 or 1( the fact is, it doesn't terminate). Explain the apparent paradox that this alternating pattern still adds to 10/99.

2. Originally Posted by lifeunderwater
Use the sum of an infinite geometric series to prove that o.101010...=10/99. Notice how this decimal expression alternates between 0 and 1 indefinitely, so essentially we do not know whether it terminates on 0 or 1( the fact is, it doesn't terminate). Explain the apparent paradox that this alternating pattern still adds to 10/99.

Should be $\sum\limits_{n = 1}^\infty {\left( {0.1} \right)^{2n - 1} } = 0.1 + 0.001 + 0.00001 + \cdots$?