Let an=(2 n)/(3 n+1) (a) Determine whether {an} is convergent. (b) Determine whether Σn=1^∞ an i

Name: Problem 9, Chapter 11: Calculus Early Transcendentals
Uploaded: 2021-08-23T20:35:14-08:00
Duration: 1 min 20 s
Channel: Lucas Finney
Description: Let an=(2 n)/(3 n+1) (a) Determine whether {an} is convergent. (b) Determine whether Σn=1^∞ an is convergent.

step 1 For this problem, we are asked to let A .N equal 2N over 3N plus 1. For the first part of the pr

Question

Let $$a_{n}=\frac{2 n}{3 n+1}$$ (a) Determine whether $\left\{a_{n}\right\}$ is convergent. (b) Determine whether $\Sigma_{n=1}^{\infty} a_{n}$ is convergent.

   Let $$a_{n}=\frac{2 n}{3 n+1}$$
(a) Determine whether $\left\{a_{n}\right\}$ is convergent.
(b) Determine whether $\Sigma_{n=1}^{\infty} a_{n}$ is convergent.

Calculus Early Transcendentals

James Stewart 6th Edition

Chapter 11, Problem 9

Instant Answer

Solved by Expert Lucas Finney

08/23/2021

Step 1

This can be done by dividing the numerator and the denominator by $n$ to get $\frac{2}{3 + \frac{1}{n}}$. Show more…

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Let $$a_{n}=\frac{2 n}{3 n+1}$$ (a) Determine whether $\left\{a_{n}\right\}$ is convergent. (b) Determine whether $\Sigma_{n=1}^{\infty} a_{n}$ is convergent.

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Key Concepts

Divergence Test for Series

For an infinite series to converge, the individual terms must approach zero as the index increases. This is known as the nth-term test or divergence test. If the limit of the terms is nonzero, the series diverges, which is a fundamental test for determining series convergence.

Limit of Rational Sequences

When analyzing the convergence of sequences composed of rational functions (fractions with polynomials in the numerator and denominator), it is useful to compare the degrees of the polynomials. In such cases, finding the limit as the index grows involves dividing the numerator and denominator by the highest power of n. This technique shows that the sequence converges to the ratio of the leading coefficients if the degrees are the same.

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