Sequences and Series

Convergence and Divergence

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A sequence is an ordered list of numbers, and a series is the sum of the terms in a sequence. These ideas matter because they describe patterns that continue step by step and help mathematicians model growth, decay, approximation, and infinite processes. In calculus, sequences and series connect algebraic patterns to limits, which makes them essential for understanding advanced functions and approximations.

A sequence approaches a limit $L$ if the terms get arbitrarily close to $L$ as $n$ becomes large. A series is written as a $\sum_{n=1}^{\infty}$ , and its behavior depends on the partial sums. If the partial sums approach a finite number, the series converges; otherwise it diverges. Important tools include geometric series formulas, convergence tests, and the idea that an infinite sum can still have a finite value.

Key Facts

A sequence converges to $L$ if $\lim_{n\to\infty} a_n = L$ .
The $n$ th partial sum of a series is $S_N = a_1 + a_2 + \ldots + a_N$ .
An infinite series converges if lim(N to infinity) S_N exists and is finite.
Geometric sequence: $a_n = a_1 \times r^{n-1}$ .
Infinite geometric series: $\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}$ , for $|r| < 1$ .
If $\sum_{n=1}^{\infty} a_n$ converges, then $\lim_{n\to\infty} a_n = 0$ .

Vocabulary

Sequence: A sequence is an ordered list of terms, usually written as a_1, a_2, a_3, and so on.
Series: A series is the sum of the terms of a sequence.
Limit: A limit is the value that a sequence or partial sum approaches as the index grows without bound.
Partial sum: A partial sum is the sum of the first N terms of a series.
Convergence: Convergence means that a sequence or series approaches a finite value as the number of terms increases.

Common Mistakes to Avoid

Assuming a_n to 0 guarantees that the series sum of a_n converges, which is wrong because many series with terms going to zero still diverge, such as the harmonic series.
Using the geometric series formula when |r| >= 1, which is wrong because the infinite geometric series only converges for |r| < 1.
Confusing the sequence a_n with the series sum of a_n, which is wrong because one is a list of terms and the other is the accumulation of those terms.
Stopping after checking only a few terms numerically, which is wrong because convergence depends on long term behavior as n becomes very large.

Practice Questions

1 Find the limit of the sequence a_n = (3n + 2) / (n + 5) as n approaches infinity.
2 Determine whether the series $\sum_{n=0}^{\infty} 5\left(\frac{1}{3}\right)^n$ converges, and if it does, find its sum.
3 A sequence has terms that get closer and closer to 4 as n increases. Explain what this means in terms of the limit, and state whether this alone tells you that the series sum of a_n converges.