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Chapter 10: Problem 29

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{4^{n}}{3^{n}+1} $$

Short Answer

Expert verified
The series \( \sum_{n=0}^{\infty} \frac{4^{n}}{3^{n}+1} \) diverges.

Step by step solution

01

Write out the Ratio Test formula

The Ratio Test formula is given by - \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \] If the limit is less than 1, the series converges, if it's more than 1, it diverges. If it equals 1, the test is inconclusive.
02

Calculate the (n+1)th term

The (n+1)th term for this series is calculated by replacing n with n+1 in the series. This yields \(\frac{4^{n+1}}{3^{n+1}+1}\)
03

Calculate the ratio of (n+1)th term to nth term

Replacing \(a_{n+1}\) with \(\frac{4^{n+1}}{3^{n+1}+1}\) and \(a_n\) with \(\frac{4^{n}}{3^{n}+1}\), we say the expression for the Ratio Test becomes \[ \lim_{n \to \infty} \left| \frac{\frac{4^{n+1}}{3^{n+1}+1}}{\frac{4^{n}}{3^{n}+1}} \right| = \lim_{n \to \infty} \left| \frac{4(4^{n})}{3(4^n)+4} \right| = \lim_{n \to \infty} \left| \frac{4}{1+\frac{4}{3(4^n)}} \right| \]
04

Calculate the limit

Now we just need to compute the limit. We know that \(\lim_{n \to \infty} \frac{4}{3(4^n)} = 0\). Replacing this into our expression gives \(\lim_{n \to \infty} \left| \frac{4}{1+0} \right| = 4\)
05

Apply the conclusion of the Ratio Test

The limit, 4, is greater than 1 which implies by the Ratio Test that the series diverges.

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Most popular questions from this chapter

Verify that the infinite series diverges. $$ \sum_{n=1}^{\infty} \frac{n}{2 n+3}=\frac{1}{5}+\frac{2}{7}+\frac{3}{9}+\frac{4}{11}+\cdots $$

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