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Chapter 9: Problem 54

Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$\sum_{n=1}^{\infty}\left(\frac{2 \pi}{3}\right)^{n}$$

Short Answer

Expert verified
The series diverges since \( |\frac{2\pi}{3}| > 1 \). The test used to determine this was the Ratio Test.

Step by step solution

01

Identify type of Series

Recognize that the series is geometric because it has a constant ratio between successive terms. In this case, the ratio (r) is \(\frac{2\pi}{3}\)
02

Apply Ratio Test

Calculate the absolute value of r: \( |\frac{2\pi}{3}| \).
03

Determine Convergence or Divergence

Compare \( |\frac{2\pi}{3}| \) with 1. If \( |\frac{2\pi}{3}| < 1 \), the series converges. If \( |\frac{2\pi}{3}| >= 1 \), the series diverges.

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