AP Calculus BC Unit 10 Study Guide: Infinite Sequences, Series, Convergence Tests, and Power Series

Sequence

An ordered list of numbers; can be viewed as a function with input a positive integer n and output a_n.

nth term (a_n)

The value of a sequence at term number n.

Limit of a sequence

A number L such that an approaches L as n→∞; written lim{n→∞} a_n = L.

Convergent sequence

A sequence whose terms approach a single finite limit as n→∞.

Divergent sequence

A sequence that does not approach a single finite number (may grow without bound or oscillate).

Oscillating sequence

A divergent sequence that alternates between values without settling to one limit (e.g., (-1)^n).

Bounded sequence

A sequence whose terms stay between two fixed numbers; bounded does not necessarily imply convergence.

Dominant-term reasoning

A limit technique where the highest-power terms control behavior as n→∞ (often used for rational expressions).

Epsilon–N definition (for sequences)

an→L if for every ε>0 there exists an integer N such that |an−L|<ε for all n≥N.

Infinite series

The sum of infinitely many terms from a sequence, written ∑{n=1}^{∞} an.

Term of a series

An individual addend an in the series ∑ an.

Partial sum (S_n)

The sum of the first n terms of a series: Sn = ∑{k=1}^{n} a_k.

Convergence of a series

A series converges if its partial sums S_n approach a finite limit S as n→∞.

Divergence of a series

A series diverges if its partial sums do not approach a finite limit (so it has no sum).

nth-term test for divergence

If lim{n→∞} an ≠ 0 or does not exist, then ∑ a_n diverges.

Necessary but not sufficient condition (terms→0)

For a convergent series, an must go to 0, but an→0 alone does not guarantee the series converges.

Harmonic series

The series ∑_{n=1}^{∞} 1/n; it diverges even though terms go to 0.

Telescoping series

A series whose partial sums simplify by cancellation of most terms, allowing direct computation of S_n and its limit.

Geometric series

A series of the form ∑ ar^n (or ∑ ar^{n-1}) with constant ratio r between successive terms.

Common ratio (r)

In a geometric series, the factor multiplying each term to get the next term.

Geometric series convergence criterion

A geometric series converges if |r|<1 and diverges if |r|≥1.

Sum of an infinite geometric series

If |r|<1, then ∑_{n=0}^{∞} ar^n = a/(1−r).

Finite geometric sum formula

For Sn = a + ar + … + ar^{n−1}, Sn = a(1−r^n)/(1−r).

p-series

A series of the form ∑_{n=1}^{∞} 1/n^p; converges if p>1 and diverges if p≤1.

Integral Test

If an=f(n) where f is continuous, positive, and decreasing (eventually), then ∑ an and ∫ f(x)dx either both converge or both diverge.

Improper integral (to infinity)

An integral with an infinite upper limit, such as ∫_{N}^{∞} f(x) dx, used to test convergence.

Remainder (R_n) for a series

The tail after n terms: Rn = ∑{k=n+1}^{∞} ak = S − Sn (if the series sum S exists).

Integral Test remainder bounds

If f is positive and decreasing and an=f(n), then ∫{n+1}^{∞} f(x)dx ≤ Rn ≤ ∫{n}^{∞} f(x)dx.

Direct Comparison Test

For nonnegative terms: compare an to bn to conclude convergence (an≤bn, ∑bn convergent) or divergence (an≥bn, ∑bn divergent), eventually.

Limit Comparison Test

If an>0, bn>0, and lim{n→∞} (an/bn)=c with 0

Alternating series

A series whose terms change sign, often written ∑ (-1)^{n-1} bn with bn≥0.

Alternating Series Test (Leibniz Test)

An alternating series ∑ (-1)^{n-1} bn converges if bn decreases eventually and lim{n→∞} bn = 0.

Alternating Series Estimation Theorem

If an alternating series converges by the AST, then the error after n terms satisfies |Rn| ≤ b{n+1}.

Absolute convergence

A series ∑ an is absolutely convergent if ∑ |an| converges.

Conditional convergence

A series that converges, but does not converge absolutely (∑ an converges while ∑|an| diverges).

Absolute Convergence Theorem

If ∑|an| converges, then ∑an converges.

Ratio Test

Compute L = lim{n→∞} |a{n+1}/a_n|: if L

Root Test

Compute L = lim{n→∞} (|an|)^{1/n}: if L

Inconclusive test result

When a convergence test (like ratio/root) yields a value (often 1) that does not determine convergence or divergence, requiring a different method.

Power series

A series of the form ∑{n=0}^{∞} an (x−c)^n, an “infinite polynomial” centered at c.

Center of a power series (c)

The value of x about which a power series is written in powers of (x−c).

Radius of convergence (R)

A number R such that a power series converges for |x−c|R (endpoints must be tested separately).

Interval of convergence

The set of x-values for which a power series converges, including any endpoints that pass separate testing.

Endpoint testing (power series)

Checking convergence at x=c−R and x=c+R because behavior at endpoints may differ from interior points.

Term-by-term differentiation (power series)

Within the interval of convergence, d/dx of ∑ an(x−c)^n equals ∑ n an (x−c)^{n−1}.

Term-by-term integration (power series)

Within the interval of convergence, ∫ ∑ an(x−c)^n dx equals C + ∑ an/(n+1)^{n+1}.

Taylor polynomial (degree n)

Pn(x)=∑{k=0}^{n} f^{(k)}(a)/k! · (x−a)^k, approximating f near x=a.

Taylor series

The infinite extension ∑_{k=0}^{∞} f^{(k)}(a)/k! · (x−a)^k (when it converges).

Maclaurin series

A Taylor series centered at 0: ∑_{k=0}^{∞} f^{(k)}(0)/k! · x^k.

Lagrange error bound (Taylor remainder)

If |f^{(n+1)}(t)|≤M between a and x, then |R_n(x)| ≤ M|x−a|^{n+1}/(n+1)!.