Chapter 8: Q. 9 (page 669)

Complete Example 4 by showing that the power series $\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k}$ diverges when $x = 2$ .

Short Answer

Expert verified

Ans: By the divergence test, the power series $\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k}$ diverges when $x = 2$

Step by step solution

Step 1. Given information.

given,

$\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k}$

Step 2. We evaluate the series when x=2

So,

$\begin{aligned} \sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k} = \sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} [2 (2) - 3]^{k} \\ = \sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} \end{aligned}$

Therefore, $k \to \infty$ , $\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} \to \infty$

Step 3. Thus,

Thus, by the divergence test, the power series $\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k}$ diverges when $x = 2$

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Complete Example 4 by showing that the power series $\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k}$ diverges when $x = 2$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. We evaluate the series when x=2

Step 3. Thus,

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Complete Example 4 by showing that the power series ∑k=0∞ (−1)kk3k+1(2x−3)kdiverges when x=2.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. We evaluate the series when x=2

Step 3. Thus,

One App. One Place for Learning.

Most popular questions from this chapter

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Complete Example 4 by showing that the power series $\sum_{k = 0}^{\infty} (- 1)^{k} \frac{k}{3 k + 1} (2 x - 3)^{k}$ diverges when $x = 2$ .