Unit 10: Infinite Sequences and Series

Sequence

An infinite ordered list of numbers, often written {an}, where an is the nth term.

nth term (a_n)

The term of a sequence corresponding to the integer index n.

Sequence as a function

A sequence viewed formally as a function whose domain is the positive integers (n = 1, 2, 3, …).

Limit of a sequence

A number $L$ such that $a_n$ approaches $L$ as $n \to \infty$; written $\lim_{n\to\infty} a_n = L$.

Convergent sequence

A sequence whose limit exists and is a finite real number.

Divergent sequence

A sequence whose limit does not exist or is infinite ($\to \infty$ or $\to -\infty$).

Oscillating sequence

A divergent sequence whose terms bounce between values and never settle to one limit (e.g., a_n = (−1)^n).

Epsilon (ε) definition of sequence limit

A sequence has limit $L$ if for every \textstyle\big\backepsilon > 0, there exists an integer $N$ such that |a_n - L| < \textstyle\big\backepsilon for all $n \not< N$.

Dominant-term reasoning (for sequence limits)

A method for limits of rational expressions in n that compares highest powers of n to find the limiting behavior.

Squeeze Theorem (for sequences)

If an is trapped between two sequences with the same limit, then an has that limit as well.

Necessary condition for series convergence

If the series $\sum a_n$ converges, then $\lim_{n\to\infty} a_n = 0$.

Converse error (common mistake)

The false claim that if an → 0, then the series Σ an must converge.

Infinite series

An expression adding infinitely many terms, written a1 + a2 + a3 + 7 ext{ or } \\sum_{n=1}^{\infty} a_n.

Sigma notation (Σ)

Compact notation for a sum, such as $\sum_{n=1}^{\infty} a_n$ for an infinite series.

Partial sum (S_N)

The finite sum of the first $N$ terms of a series: S_N = \textstyle \bigsum_{n=1}^{N} a_n.

Sequence of partial sums

The sequence {S_N} formed by the partial sums; its limit determines whether the series converges.

Sum of a series (S)

If $\lim_{N\to\infty} S_N = S$ (finite), then the series converges and S is defined as its sum.

Series convergence (definition via partial sums)

A series $\sum a_n$ converges exactly when its partial sums $S_N$ converge to a finite limit.

nth-term test for divergence

If $\lim_{n\to\infty} a_n \neq 0$ or does not exist, then $\sum a_n$ diverges.

One-way nature of the nth-term test

The nth-term test can prove divergence only; if a_n → 0, the test gives no conclusion.

Geometric series

A series with constant ratio $r$ between successive terms; common forms include \textstyle \bigsum_{n=0}^{\textstyle\biginfty} a r^n.

Common ratio (r)

In a geometric series, the constant factor relating terms (each term is multiplied by r to get the next).

Geometric series convergence rule

A geometric series converges if and only if $|r| < 1$; it diverges if $|r| \not< 1$.

Finite geometric sum formula

For first term $a$ and ratio $r$, the sum of first $n$ terms is $S_n = \frac{a(1 - r^n)}{1 - r}$, for $r \neq 1$.

Infinite geometric sum formula

If $|r| < 1$, then $\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}$.

Telescoping series

A series that simplifies via cancellation when writing partial sums, leaving only a few surviving terms.

Benchmark series

A commonly known series (like harmonic or p-series) used as a comparison reference for convergence.

Harmonic series

The series \textstyle \bigsum_{n=1}^{\textstyle \biginfty} \frac{1}{n}, which diverges even though its terms go to 0.

p-series

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$ with a complete convergence rule based on p.

p-series convergence rule

\textstyle \bigsum \frac{1}{n^p} converges if $p > 1$ and diverges if $p \not> 1$.

Positive-term series property

If all terms are positive, partial sums are increasing; the series either levels off (converges) or grows without bound (diverges).

Direct comparison test

If 0 \le an \le bn eventually, then (i) \sum bn convergent \Rightarrow \sum an convergent; (ii) \sum an divergent \Rightarrow \sum bn divergent.

Inequality with denominators (comparison tip)

When denominators are larger, fractions are smaller; flipping this is a common comparison mistake.

Limit comparison test

For $a_n, b_n > 0$, compute L = \textstyle\biglim \frac{a_n}{b_n}. If $0 < L < \infty$, then \textstyle \bigsum a_n and \textstyle \bigsum b_n either both converge or both diverge.

Choosing b_n in limit comparison

Pick bn to match the dominant large-n behavior of an (often like 1/n^p).

Integral test

If $f$ is positive, continuous, and decreasing on $[1,\infty)$ and $a_n = f(n)$, then $\sum a_n$ and $\int_1^{\infty} f(x) \, dx$ either both converge or both diverge.

Conditions for the integral test

To apply the integral test, f(x) must be positive, continuous, and decreasing for x ≥ 1.

Ratio test

For $a_n > 0$, let $L = \lim \frac{a_{n+1}}{a_n}$. If $L < 1$ converge; if $L > 1$ or $\infty$ diverge; if $L = 1$ inconclusive.

Ratio test inconclusive case

When L = 1 in the ratio test, you cannot conclude convergence or divergence from the test.

Root test

Compute L = \textstyle\biglim \root{n}{a_n}. If $L < 1$ converge; if $L > 1$ diverge; if $L = 1$ inconclusive.

Root test inconclusive case

When L = 1 in the root test, the test does not decide convergence.

Alternating series

A series whose terms alternate signs, often written $\sum (-1)^{n+1} b_n$ with $b_n > 0$.

Alternating series test (Leibniz test)

An alternating series converges if $b_n$ is decreasing (eventually) and $\lim b_n = 0$ (with sign alternation present).

Absolute convergence

A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges (which guarantees $\sum a_n$ converges).

Conditional convergence

A series $\sum a_n$ converges, but the absolute value series $\sum |a_n|$ diverges.

Absolute Convergence Theorem

If \sum |an| converges, then \sum an converges.

Alternating series remainder (error) bound

If an alternating series converges by the alternating series test, then $|R_N| = |S - S_N| \leq b_{N+1}$ (the next term’s magnitude).

Power series

A series of the form \textstyle \bigsum_{n=0}^{\textstyle \biginfty} c_n (x - a)^n, centered at $x = a$.

Geometric series

A series with constant ratio $r$ between successive terms; common forms include $\sum_{n=0}^{\infty} a r^n$.

Interval of convergence

The set of x-values where a power series converges; endpoints (where |x − a| = R) must be tested separately.