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

step 1 Okay, so the sequence, A .N is 2n over 3N plus 1. So the first thing that we need to determine i

Let an=(2 n)/(3 n+1) (a) Determine whether {an} is...

Let $a_{n}=\frac{2 n}{3 n+1}$
\begin{equation}
\begin{array}{l}{\text { (a) Determine whether }\left\{a_{n}\right\} \text { is convergent. }} \\ {\text { (b) Determine whether } \sum_{n=1}^{\infty} a_{n} \text { is convergent. }}\end{array}
\end{equation}

Key Concepts

Sequence Convergence

This concept involves the study of whether the terms of a sequence approach a specific value as the index increases without bound. A sequence is said to be convergent if the terms eventually become arbitrarily close to a single number, known as the limit. In the context of rational functions of the form (polynomial)/(polynomial), the behavior as the index grows large is often determined by comparing the highest power terms in the numerator and the denominator.

Dominant Term Analysis in Rational Functions

When evaluating the limit of a sequence defined by a rational function, it is common to focus on the terms with the highest powers in both the numerator and the denominator. This analysis simplifies the sequence and helps in determining its long-term behavior by mitigating the contributions of lower order terms. This technique is particularly useful for sequences where both the numerator and denominator grow without bound.

Infinite Series Convergence

This concept refers to determining whether the sum of an infinite sequence of terms is finite, that is, whether the series converges. A necessary condition for the convergence of an infinite series is that the individual term of the series tends to zero as the index approaches infinity. However, even if this condition is satisfied, further tests are often required to confirm convergence.

Divergence Test for Infinite Series

The divergence test, also known as the nth-term test, states that if the limit of the sequence of terms in an infinite series does not equal zero, then the series must diverge. This test is used as a preliminary check for series convergence and is a fundamental tool in the analysis of infinite series.

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