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Chapter 10: Problem 21

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} n\left(\frac{3}{4}\right)^{n} $$

Short Answer

Expert verified
The series \(\sum_{n=1}^{\infty} n(\frac{3}{4})^{n}\) is convergent, since the limit calculated through the Ratio Test is less than 1.

Step by step solution

01

Write down the formula for the Ratio Test

The Ratio Test states that for a series \(\sum_{n=1}^{\infty} a_n\), we consider \(L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\). If \(L < 1\), the series is absolutely convergent. If \(L > 1\), the series is divergent. If \(L = 1\), the test is inconclusive.
02

Apply the Ratio Test to the given series

For the given series \(\sum_{n=1}^{\infty} n(\frac{3}{4})^{n}\), we need to compute the limit \(L = \lim_{n \to \infty} \left|\frac{(n+1)(\frac{3}{4})^{n+1}}{n(\frac{3}{4})^{n}}\right|\). Compute this limit.
03

Simplify the fraction

After simplifying, the equation becomes \(L = \lim_{n \to \infty} \left|\frac{n+1}{n}脳\frac{3}{4}\right|\). Distribute the \(\frac{3}{4}\) to the components inside the absolute value, and compute this limit.
04

Evaluate the limit

As \(n \to \infty\), \((n+1)/n\) tends to 1. Thus, the limit becomes \(L = \left|\frac{3}{4}\right|\), which is less than 1.

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Most popular questions from this chapter

Use a symbolic algebra utility to find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 2\left(-\frac{2}{3}\right)^{n}=2-\frac{4}{3}+\frac{8}{9}-\frac{16}{27}+\cdots $$

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=0}^{\infty} n ! $$

Use a symbolic algebra utility to find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 2\left(\frac{2}{3}\right)^{n}=2+\frac{4}{3}+\frac{8}{9}+\frac{16}{27}+\cdots $$

Write the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{3^{n}}{2^{n-1}}=3-\frac{9}{2}+\frac{27}{4}-\frac{81}{8}+\frac{243}{16}-\cdots $$

The random variable \(n\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n)\). Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(0)+P(1)+P(2)+P(3)+\cdots=1\) $$ P(n)=\frac{1}{2}\left(\frac{1}{2}\right)^{n} $$
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