Final answer:
The alternating sum S = 1 - 1/2 + 1/4 - 1/8 + ... + 1/65536 is a finite geometric series, which can be calculated using the sum formula for geometric sequences, resulting in a simple fraction.
Step-by-step explanation:
To express the alternating sum S = 1 - 1/2 + 1/4 - 1/8 + ... + 1/65536 as a simple fraction, we recognize that it is a finite geometric series with a common ratio of -1/2. The first term, a, is 1, and each subsequent term is half of the previous term and alternates in sign. We use the formula for the sum of a finite geometric series:
S = a(1 - r^n) / (1 - r),
where n is the number of terms, a is the first term and r is the common ratio. Here, r = -1/2, and n can be found by noting that the last term, 1/65536, is 2^(-16), so there are 17 terms (since the series starts with an exponent of 0).
Plugging in the values we get:
S = 1(1 - (-1/2)^17) / (1 - (-1/2)) = (1 - (-1/2)^17) / (3/2).
Simplifying the above expression, we have the sum S as a simple fraction. The final step is to calculate (-1/2)^17 and subtract it from 1, which is straightforward since (-1/2)^17 is a very small number.