Sequences and Series

Convergence, Tests & Sums

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Sequences and series are tools for describing what happens when a process continues step by step forever. A sequence lists numbers in order, while a series adds those numbers together. Convergence tells us whether the terms of a sequence approach a single value or whether the partial sums of a series settle toward a finite number. This idea matters because infinite processes appear in calculus, physics, engineering, and numerical computation.

To study convergence, mathematicians compare terms or partial sums to a limit and ask whether they get arbitrarily close after enough steps. For sequences, the key question is whether $a_n$ approaches $L$ as $n$ becomes large. For series, we examine the sequence of partial sums $S_n = a_1 + a_2 + \ldots + a_n$ and test whether $S_n$ approaches a finite value. Different convergence tests, such as the geometric series rule, comparison test, ratio test, and integral test, help determine what happens without adding infinitely many terms directly.

Key Facts

A sequence converges to L if for every epsilon > 0, there exists N such that n > N implies |a_n - L| < epsilon.
A series $\sum a_n$ converges if its partial sums $S_n = \sum_{k=1}^{n} a_k$ approach a finite limit.
If $\sum a_n$ converges, then $a_n \to 0$ . The converse is false.
Geometric series: $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1 - r}$ when $|r| < 1$ .
p-series: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$ .
Ratio test: if $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = r$ , then the series converges for $r < 1$ and diverges for $r > 1$ .

Vocabulary

Sequence: A sequence is an ordered list of numbers written as a_1, a_2, a_3, and so on.
Series: A series is the sum of the terms of a sequence, often written as a_1 + a_2 + a_3 + ...
Limit: A limit is the value that sequence terms or partial sums approach as the index grows without bound.
Partial sum: A partial sum is the sum of the first n terms of a series, usually denoted S_n.
Convergence test: A convergence test is a method used to decide whether an infinite series converges or diverges.

Common Mistakes to Avoid

Assuming $a_n \to 0$ guarantees that $\sum a_n$ converges, but this is wrong because many series, such as $\sum \frac{1}{n}$ , have terms going to zero and still diverge.
Applying a sequence limit directly to a series, which is wrong because a series must be studied through its partial sums rather than through the terms alone.
Using the geometric series formula when |r| >= 1, which is wrong because the infinite geometric series only converges for |r| < 1.
Stopping after the ratio test gives r = 1, which is wrong because the ratio test is inconclusive there and another test must be used.

Practice Questions

1 Determine whether the sequence a_n = (3n + 1)/(n + 5) converges, and if it does, find its limit.
2 Decide whether the series $\sum_{n=1}^{\infty} 4\left(\frac{1}{3}\right)^n$ converges, and if so, find its sum.
3 A student says that because the terms of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... get very small, the series must converge. Explain clearly why this reasoning is not sufficient.

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