Question

For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$$

Short answer

Answer: The infinite geometric series converges to a sum of 2.

Step-by-step solution

Identify the infinite series

The given series is an infinite geometric series with the first term, a, equal to 1, and the common ratio, r, equal to 1/2. The series can be represented as: $$\sum_{n=0}^{\infty} a \cdot r^n$$, which in this case is: $$\sum_{n=0}^{\infty} 1 \cdot \left(\frac{1}{2}\right)^n$$

Calculate the first partial sum

To find the first partial sum, S1, add the first term of the series: $$S_1 = 1$$

Calculate the second partial sum

To find the second partial sum, S2, add the first and second terms of the series: $$S_2 = 1 + \frac{1}{2} = \frac{3}{2}$$

Calculate the third partial sum

To find the third partial sum, S3, add the first, second, and third terms of the series: $$S_3 = 1 + \frac{1}{2} + \frac{1}{4} = \frac{7}{4}$$

Calculate the fourth partial sum

To find the fourth partial sum, S4, add the first, second, third, and fourth terms of the series: $$S_4 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8}$$

Make a conjecture about the value of the infinite series

To make a conjecture about the value of the infinite series, observe the pattern of the partial sums. We can see that the partial sums are getting closer to a specific value: $$1, \frac{3}{2}, \frac{7}{4}, \frac{15}{8}, \dots$$ The denominators are increasing powers of 2 and the numerators are 1 less than the next power of 2. This suggests that the infinite series converges to a value close to but less than 2. In fact, it converges to 2 as the partial sums become: $$1, \frac{3}{2}, \frac{7}{4}, \frac{15}{8}, \dots$$ which can be rewritten as: $$\frac{2}{2},\frac{4}{2},\frac{8}{4},\frac{16}{8},\dots$$ The numerators are increasing powers of 2 (excluding the first term), and the denominators are powers of 2 as well. As the value of n approaches infinity, the difference between the numerator and denominator in each fraction converges to 0, thus the series converges to 2. Therefore, we can conjecture that the infinite series converges to a sum of 2.