Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.
The given repeating decimal as a geometric series is and as the quotient of two integers reduced to lowest terms is
The given repeating decimal is
The repeating decimal as a geometric series can be expressed as
The given repeating decimal as the quotient of two integers reduced to the lowest terms can be expressed as
Let be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to . (Hint: Argue that if you add up some finite number of the terms of , the sum will be greater than . Then argue that, by adding in some other finite number of the terms of
, you can get the sum to be less than . By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to .)
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