Chapter 7: Q 59. (page 615)

Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges to $\frac{64}{3}$ .

Step by step solution

Step 1. Given information.

Given a series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ .

Step 2. Find if the series converges or not.

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

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

In the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ it can be seen that $c = \frac{80}{3}$ .

Every term after that is $- \frac{1}{4}$ times the previous term.

It follows that $r = - \frac{1}{4}$ .

Since $r = - \frac{1}{4}$ , the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ 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 localid="1648982757515" $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges to localid="1648982761274" $\frac{\frac{80}{3}}{1 - (\frac{- 1}{4})}$ , that is $\frac{64}{3}$ .

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Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ 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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Determine whether the series 803-203+53-512+…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.

One App. One Place for Learning.

Most popular questions from this chapter

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Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges or diverges. Give the sum of the convergent series.