Final answer:
The formula for the sum of the series 1/2 + 1/4 + 1/8 + ⋯ + 1/(2n) is 1 - 1/2^n. This can be derived by examining the pattern of the terms and using the formula for the sum of a geometric series.
Step-by-step explanation:
The formula for the sum of the series 1/2 + 1/4 + 1/8 + ⋯ + 1/(2n) is 1 - 1/2^n. To derive this formula, let's examine the pattern:
- The first term is 1/2.
- The second term is 1/4, which is 1/2^2.
- The third term is 1/8, which is 1/2^3.
- ...
- The nth term is 1/(2n), which is 1/2^n.
To find the sum, we can use the formula for the sum of a geometric series: S = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 1/2, r = 1/2, and n is the number of terms. Plugging in these values gives us S = (1/2)(1 - (1/2)^n) / (1 - 1/2).
We simplify this expression as S = 1 - 1/2^n, which is the formula for the sum of the series 1/2 + 1/4 + 1/8 + ⋯ + 1/(2n).