Question

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \frac{3}{10^{k}}$$

Short answer

Question: Write the first four terms of the sequence of partial sums for the series $$\sum_{k=1}^{\infty} \frac{3}{10^{k}}$$ and estimate the limit of the sequence of partial sums or state that it does not exist. Answer: The first four terms of the sequence of partial sums are: \(S_1 = 0.3\), \(S_2 = 0.33\), \(S_3 = 0.333\), and \(S_4 = 0.3333\). The limit of the sequence of partial sums is estimated to be \(S = 0.\overline{3}\).

Step-by-step solution

1. Identify the general term of the given series

The given series is $$\sum_{k=1}^{\infty} \frac{3}{10^{k}}$$, where the general term is given by \(a_k = \frac{3}{10^k}\) for \(k \in \mathbb{N}\).

2. Compute the first four terms of the partial sums

The sequence of partial sums is given by: \(S_1 = a_1 = \frac{3}{10^1}\) \(S_2 = a_1 + a_2 = \frac{3}{10^1} + \frac{3}{10^2}\) \(S_3 = a_1 + a_2 + a_3 = \frac{3}{10^1} + \frac{3}{10^2} + \frac{3}{10^3}\) \(S_4 = a_1 + a_2 + a_3 + a_4 = \frac{3}{10^1} + \frac{3}{10^2} + \frac{3}{10^3} + \frac{3}{10^4}\) Upon simplifying, we obtain the following terms: \(S_1 = \frac{3}{10} = 0.3\) \(S_2 = 0.3 + \frac{3}{100} = 0.3 + 0.03 = 0.33\) \(S_3 = 0.33 + \frac{3}{1000} = 0.33 + 0.003 = 0.333\) \(S_4 = 0.333 + \frac{3}{10000} = 0.333 + 0.0003 = 0.3333\)

3. Estimate the limit of the sequence of partial sums

The given series is a geometric series with the common ratio \(r = \frac{1}{10}\) and the first term \(a = \frac{3}{10}\). Since the common ratio \(|r|\) is less than 1 (\(|r| = |\frac{1}{10}| < 1\)), the series converges, and we can use the formula for the sum of an infinite geometric series to estimate the limit of the sequence of partial sums: $$S = \frac{a}{1-r} = \frac{\frac{3}{10}}{1- \frac{1}{10}} = \frac{\frac{3}{10}}{\frac{9}{10}} = \frac{1}{3} = 0.\overline{3}$$ The limit of the sequence of partial sums is estimated to be \(S = 0.\overline{3}\).