Determining Sequences in Calculus

To successfully solve questions about determining sequences in calculus, a student must focus on finding the limit of an ordered list of numbers as the index approaches infinity. Unlike series, which accumulate sums, a sequence is simply a function whose domain is the set of positive integers. The core objective is to analyze whether the terms eventually stabilize toward a specific, finite value, which indicates convergence, or if they grow without bound or oscillate, which indicates divergence.

The first practical step is to write out the general term of the sequence, typically denoted as `a_n`. If the problem provides the first few numbers of a sequence instead of an explicit formula, the student must analyze the patterns to deduce the algebraic expression for `a_n` as a function of `n`. Once the general term is established, determining the behavior of the sequence mathematically translates to evaluating the limit `\lim_{n \to \infty} a_n` using standard calculus limit laws.

When evaluating these limits, students can utilize powerful tools from single-variable calculus, such as L'Hôpital's Rule and the Squeeze Theorem. Because `n` only takes integer values, you can conceptually replace `n` with a continuous real variable `x` to safely apply derivatives. Additionally, if a sequence is found to be both bounded and strictly monotonic (constantly increasing or constantly decreasing), the Monotonic Sequence Theorem guarantees that the sequence converges, even if the exact limit is difficult to calculate directly.

Let us look at a practical example: determine whether the sequence defined by the general term `a_n = \frac{3n^2 + 5}{2n^2 + n}` converges or diverges, and find its limit if it converges.

To find the behavior, we set up the limit as `n` approaches infinity: `\lim_{n \to \infty} \frac{3n^2 + 5}{2n^2 + n}`. To evaluate this algebraic fraction, we divide every term in both the numerator and the denominator by the highest power of `n` present in the expression, which is `n^2`. This algebraic manipulation rewrites the sequence limit as `\lim_{n \to \infty} \frac{3 + \frac{5}{n^2}}{2 + \frac{1}{n}}`.

As `n` grows infinitely large, the fractional terms `\frac{5}{n^2}` and `\frac{1}{n}` both approach zero. Substituting these values into our expression leaves us with the simplified fraction `\frac{3 + 0}{2 + 0} = \frac{3}{2}`. Because this limit is a single, finite number, the student can confidently conclude that the sequence converges to `\frac{3}{2}`.