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Chapter 7: Problem 25

Evaluate the geometric series. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+\frac{1}{2^{80}}-\frac{1}{2^{81}} $$

Short Answer

Expert verified
The sum of the given geometric series is approximately \( S \approx \frac{2}{3}(1 - 3.122502256 \times 10^{-25}) \).

Step by step solution

01

Identify the needed formula for an alternating geometric series

For a geometric series, the sum S can be calculated using the formula: \[ S = \frac{a(1-r^n)}{1-r} \], where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. In this case, a = 1, r = -1/2, and n = 82.
02

Plug the values into the formula

Now, we will plug the values of 'a', 'r', and 'n' into the formula for the sum of the geometric series, and simplify: \[ S = \frac{1\left(1-\left(-\frac{1}{2}\right)^{82}\right)}{1-\left(-\frac{1}{2}\right)} \]
03

Simplify the expression

First, we simplify the exponent term, and then simplify the rest of the expression step by step: \[ S = \frac{1\left(1-\left(\frac{1}{2}\right)^{82}\right)}{\frac{3}{2}} \] Divide the numerator by the denominator: \[ S = \frac{2}{3}\left(1-\left(\frac{1}{2}\right)^{82}\right) \]
04

Evaluate the final expression

Now, we can evaluate the expression to find the sum of our given geometric series: \[ S = \frac{2}{3}\left(1-\left(\frac{1}{2}\right)^{82}\right) \] We can now use a calculator to find the exact value of this sum: $$ S \approx \frac{2}{3}(1 - 3.122502256 \times 10^{-25}) $$

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