Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\left\\{\frac{4 n^{2}-n+5}{3 n^{2}+1}\right\\}$$

Short answer

The sequence converges and the limit is \( \frac{4}{3} \).

Step-by-step solution

Identify the Sequence

The sequence given is \( \left\{ a_{n} \right\} = \left\{ \frac{4n^2 - n + 5}{3n^2 + 1} \right\} \)

Analyze Highest Powers of n

Observe that the highest power of \( n \) in both the numerator and the denominator is \( n^2 \). This indicates that the leading terms determine the behavior of the sequence as \( n \) approaches infinity.

Simplify the Expression

Divide every term in the numerator and the denominator by \( n^2 \), the highest power of \( n \):\[a_{n} = \frac{4 - \frac{1}{n} + \frac{5}{n^2}}{3 + \frac{1}{n^2}}\]

Determine the Limit as n Approaches Infinity

As \( n \to \infty \), the terms \( \frac{1}{n} \) and \( \frac{1}{n^2} \) both approach 0. Thus, the simplified expression becomes:\[a_{n} \to \frac{4}{3}\]

Conclusion on Convergence

Since the sequence converges to a constant, \( \lim_{{n \to \infty}} a_{n} = \frac{4}{3} \), it is convergent.

Key concepts

Understanding the Limit of a Sequence

A sequence can be thought of as a list of numbers generated by a specific formula for each term. The limit of a sequence is essentially the value that these numbers get closer to as you progress further into the sequence. When a sequence converges, it means that as you continue to larger and larger terms, the terms start settling closer to a particular number. This particular number is known as the limit.

To find the limit of the sequence provided, we determine the behavior of the terms as the sequence progresses towards infinity. By simplifying the expression for the sequence, we can observe how the terms interact:

• First, identify the highest power of the variable in the numerator and denominator.
• Simplify the expression to see which terms fade into insignificance as the sequence progresses.
• If it settles to a constant, that is the limit, indicating convergence.

For the sequence\[ a_{n} = \frac{4n^2 - n + 5}{3n^2 + 1} \]dividing every term by \( n^2 \) helps highlight the dominant terms. The limit then tends towards \( \frac{4}{3} \) as \( n \to \infty \), inferring convergence to this value.

What is an Infinite Series?

An infinite series sums the terms of an infinite sequence. Think of it as an endless sum where you keep adding the terms of a sequence forever. There’s a key distinction between a sequence and a series here: while a sequence lists numbers, a series adds them up.

Understanding convergence in the context of a series is slightly different from a sequence. When we talk about a series converging, we ask whether these endless sums approach a real number. To check convergence, you often analyze the sequence of partial sums:

• For a series \( \sum a_n \), you create the sequence of partial sums \( S_n = a_1 + a_2 + \cdots + a_n \).
• If \( S_n \) approaches a limit \( S \) as \( n \to \infty \), the series converges to \( S \).

The series is an essential tool in calculus, physics, and engineering, as it allows you to approximate functions and phenomena, often leading into topics like power series and Taylor series.

Tackling Calculus Problems in Convergence

Many calculus problems involve finding whether sequences or series converge, and calculating their limits. Being skilled in solving such problems can be immensely helpful, especially when tackling complex calculus challenges.

Here's how you can systematically approach convergence problems:

• Identify the sequence or series and its general term.
• Examine its terms, looking at leading terms to judge the behavior at infinity.
• Consider whether dividing by the highest power of the variable in both numerator and denominator can simplify your process.
• Employ techniques such as L'Hôpital's Rule for more complex cases when the form is indeterminate.

With practice, determining convergence becomes an intuitive process that aids in solving broader calculus problems. This understanding hinges on grasping the behavior of sequences and series as they extend towards infinity, crucial for mastering calculus.