Chapter 7: Q. 14 (page 656)

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 0}^{\infty} 3 {(\frac{- 2}{5})}^{k}$

Short Answer

Expert verified

The sum of the series is $\frac{15}{7}$ .

Step by step solution

Step 1. Given Information

The given series is $\sum_{k = 0}^{\infty} 3 {(\frac{- 2}{5})}^{k}$ .

Step 2. Determine if a geometric series

Find the ratio of the consecutive terms.

$\begin{array}{rcl} \frac{3 {(\frac{- 2}{5})}^{k + 1}}{3 {(\frac{- 2}{5})}^{k}} & = & {(\frac{- 2}{5})}^{k + 1 - k} \\ = & \frac{- 2}{5} \end{array}$

Since the ratio of the consecutive terms is same, the given series is a geometric series with a common ratio $r = \begin{array}{rcl} \frac{- 2}{5} \end{array}$ .

Step 3. Determine the convergence

Determine the first term and the common ratio.

$\begin{array}{rcl} c & = & 3 {(\frac{- 2}{5})}^{0} \\ = & 3 \\ r & = & \frac{- 2}{5} \end{array}$

If $|r| < 1$ , the series converges to $\frac{c}{1 - r}$ .

$\begin{array}{rcl} \frac{c}{1 - r} & = & \frac{3}{1 - (\frac{- 2}{5})} \\ = & \frac{3}{1 + \frac{2}{5}} \\ = & \frac{15}{7} \end{array}$

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 0}^{\infty} 3 {(\frac{- 2}{5})}^{k}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine if a geometric series

Step 3. Determine the convergence

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges. ∑k=0∞3-25k

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine if a geometric series

Step 3. Determine the convergence

One App. One Place for Learning.

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 0}^{\infty} 3 {(\frac{- 2}{5})}^{k}$