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Chapter 7: Problem 44

Use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{3^{n}}{4^{n}+5} $$

Short Answer

Expert verified
The series \(\sum_{n=0}^{\infty} \frac{3^{n}}{4^{n}+5}\) is CONVERGENT.

Step by step solution

01

Understanding the series

We are given the series \(\sum_{n=0}^{\infty} \frac{3^{n}}{4^{n}+5}\). We need to decide which series to compare it with. For this, it is beneficial to take note of the biggest term in the denominator, which is \(4^{n}\). Therefore, we will compare our series with \(\sum_{n=0}^{\infty} \frac{3^{n}}{4^{n}}\).
02

Recognizing the comparison series

The comparison series \(\sum_{n=0}^{\infty} \frac{3^{n}}{4^{n}}\) is a geometric series where the ratio \(r\), which is \(\frac{3}{4}\), is less than 1. This means by the Geometric Series Test, this comparison series is a convergent series.
03

Applying the Direct Comparison Test

The Direct Comparison test states that if 0 鈮� a鈧� 鈮� b鈧� for all n sufficiently large and \(\sum_{n=1}^{\infty} b鈧橽) is convergent, then \(\sum_{n=1}^{\infty} a鈧橽) is also convergent. In this situation, a鈧� would be \(\frac{3^{n}}{4^{n}+5}\), and b鈧� would be \(\frac{3^{n}}{4^{n}}\), and since 0 鈮� \(\frac{3^{n}}{4^{n}+5}\) 鈮� \(\frac{3^{n}}{4^{n}}\) for all n, and we have established that the series for b鈧� is convergent, this means that the given series \(\sum_{n=0}^{\infty} \frac{3^{n}}{4^{n}+5}\) is also convergent by the Direct Comparison Test.

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Most popular questions from this chapter

Modeling Data The annual sales \(a_{n}\) (in millions of dollars) for Avon Products, Inc. from 1993 through 2002 are given below as ordered pairs of the form \(\left(n, a_{n}\right),\) where \(n\) represents the year, with \(n=3\) corresponding to 1993. (Source: 2002 Avon Products, Inc. Annual Report) (3,3844),(4,4267),(5,4492),(6,4814),(7,5079) (8,5213),(9,5289),(10,5682),(11,5958),(12,6171) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n+c, \quad n=3,4, \ldots, 12\) for the data. Graphically compare the points and the model. (b) Use the model to predict sales in the year 2008 .

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