Chapter 7: Q 53. (page 615)

Determine whether the series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges to $\frac{9}{40}$ .

Step by step solution

Step 1. Given information.

Given a series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ .

Step 2. Find if the series converges or not.

The index starts with 2, rather than 0.

Note that the convergence of a series depends not upon the first few terms but only upon the tail of the series.

The standard form of geometric series is $\sum_{k = 0}^{\infty} c r^{k}$ .

Here, the series role="math" localid="1648964110412" $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ has $c = \frac{9}{25}$ and $r = \frac{- 3}{5}$ .

The geometric series converges if and only if $|r| < 1$ .

Since $r = \frac{- 3}{5}$ , it follows that the series role="math" localid="1648964322077" $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges.

Step 3. Find the value to which the series converges.

If the geometric series $\sum_{k = 0}^{\infty} c r^{k}$ converges, it converges to $\frac{c}{1 - r}$ .

So, the series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges to $\frac{\frac{9}{25}}{1 - (\frac{- 3}{5})}$ , that is $\frac{9}{40}$ .

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Determine whether the series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

Step 3. Find the value to which the series converges.

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Step by step solution

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Step 3. Find the value to which the series converges.

One App. One Place for Learning.

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Determine whether the series $\sum_{k = 2}^{\infty} {(\frac{- 3}{5})}^{k}$ converges or diverges. Give the sum of the convergent series.