Infinite Series and Sequences: Convergence, Divergence, and the Divergence Test

Vocabulary flashcards covering the definitions, formulas, and tests for convergence and divergence of sequences and infinite series.

Sequence ($a_n$)

An ordered list of individual numbers or terms, such as $2, 4, 6, 8, \dots$, generated by a general rule or term formula like $a_n = 2n$.

General Partial Sum ($s_n$)

The sum of the first $n$ terms of any sequence ($s_n = a_1 + a_2 + \dots + a_n$), representing the accumulated total before applying specific arithmetic (additive) or geometric (multiplicative) closed-form formulas.

Infinite Series

The summation of infinitely many terms of a sequence, rigorously defined as the limit of the sequence of partial sums: $\lim_{n \to \infty} s_n$.

Convergent Sequence

A sequence $a_n$ whose terms approach a specific, finite value as $n$ approaches infinity ($\lim_{n \to \infty} a_n = L$).

Divergent Sequence

A sequence $a_n$ whose terms grow or decrease without bound, or oscillate without approaching any single finite value as $n \to \infty$.

Convergent Series

A series whose infinite sequence of partial sums approaches a specific, finite constant value $S$, meaning $\lim_{n \to \infty} s_n = S$.

Divergent Series

A series whose sequence of partial sums does not settle to a finite number, such as when the partial sums approach positive or negative infinity or oscillate indefinitely.

Arithmetic Partial Sum Formula (Definition)

The method used to calculate the sum of the first $n$ terms of an arithmetic sequence, which progresses by adding a constant common difference $d$ between consecutive terms, distinct from multiplying by a common ratio.

Arithmetic Partial Sum Formula

The closed-form algebraic formula for the sum of the first $n$ terms of an arithmetic sequence with common difference $d$: $s_n = n \cdot \left(\frac{a_1 + a_n}{2}\right)$.

Geometric Partial Sum Formula (Definition)

The method used to calculate the sum of the first $n$ terms of a geometric sequence, which progresses by multiplying each successive term by a constant common ratio $r$, distinct from adding a common difference.

Geometric Partial Sum Formula

The closed-form algebraic formula for the sum of the first $n$ terms of a geometric sequence with common ratio $r \neq 1$: $s_n = \frac{a_1(1 - r^n)}{1 - r}$.

Divergence Test

A preliminary test for infinite series stating that if the sequence terms do not approach zero ($\lim_{n \to \infty} a_n \neq 0$), then the series diverges.

Inconclusive Condition of the Divergence Test

The outcome where $\lim_{n \to \infty} a_n = 0$, which does not guarantee convergence; the series may converge or diverge and requires specific structural tests (such as geometric or arithmetic analysis).

Necessary Condition for Series Convergence

The requirement that $\lim_{n \to \infty} a_n = 0$, ensuring that as $n$ becomes very large, newly added terms approach zero so the running partial sum does not continue to shift indefinitely.

Closed-Form Partial Sum of the Arithmetic Series $a_n = 2n$

The closed-form formula obtained by applying the arithmetic partial sum formula to the sequence $a_n = 2n$: $s_n = n(n + 1)$.

Divergence of the Arithmetic Series via Partial Sum Limit ($a_n = 2n$)

The evaluation showing that the arithmetic series with terms $a_n = 2n$ diverges because the limit of its arithmetic partial sum formula increases without bound: $\lim_{n \to \infty} s_n = \lim_{n \to \infty} n(n + 1) = \infty$.

Reciprocal Limit Identity at Infinity

The fundamental algebraic limit identity used to evaluate rational terms as $n$ approaches infinity: $\lim_{n \to \infty} \frac{1}{n} = 0$.

Rational Sequence Limit Evaluation ($\frac{5n + 3}{7n - 4}$)

The technique of dividing numerator and denominator by $n$, establishing that $\lim_{n \to \infty} \frac{5n + 3}{7n - 4} = \frac{5}{7}$, which proves the sequence converges to a non-zero constant.

Non-Zero Sequence Limit Divergence Principle

The principle stating that if a sequence converges to any non-zero value ($\lim_{n \to \infty} a_n \neq 0$), continuously adding that value forces the partial sums to increase or decrease indefinitely, causing the series to diverge.

Unbounded Partial Sums Criterion

The criterion stating that an infinite series is divergent whenever its sequence of partial sums does not settle to a finite value: $\lim_{n \to \infty} s_n = \pm\infty$.