unit-10 – HighFiveAP

10.1

Sequence Basics (Converge vs Diverge)

A sequence is a function with input n (a positive integer): a_n.

Definition

Converges if lim_{n→∞} a_n = L (a finite number).
Diverges if it doesn’t approach a single finite value.

Graph: Convergent vs Divergent sequences

Dots represent terms a_n at integer n. AP frequently asks you to identify convergence from a graph/table.

High-yield: Use limit laws + dominant term tricks. Example: a_n = (3n^2 + 1)/(n^2 - 5) → 3.

10.2

Series & Sigma Notation

A series adds sequence terms: Σ a_n.

Partial Sums

Define S_k = a_1 + a_2 + ... + a_k.
The series Σ a_n converges if lim_{k→∞} S_k exists.

Graph: Partial sums approaching a limit

A series converges when its partial sums settle to a finite value. This is the official definition.

10.3

Geometric Series (The One You MUST Memorize)

Σ_{n=0}^{∞} ar^n = a/(1-r) (only if |r| < 1)

How to recognize it instantly

Each term is multiplied by a constant ratio: r = a_{n+1}/a_n
Index shift doesn’t matter: start at 1 or 0, still geometric

Score-5 move: Convert repeating decimals to geometric series. Example: 0.777... = 7/10 + 7/100 + 7/1000 + ...

10.4

nth-Term Test for Divergence (Fastest Test)

If lim_{n→∞} a_n ≠ 0 (or DNE), then Σ a_n diverges.
But: if the limit equals 0, you still don’t know — you need another test.

AP habit: They will try to trick you into thinking “limit 0 ⇒ convergent.” That is FALSE. Example: harmonic series Σ 1/n diverges even though 1/n→0.

10.5

Integral Test (Bridge from Unit 8)

When you can use it

If a_n = f(n), where f(x) is continuous, positive, and decreasing for x≥1, then:

Σ a_n and ∫_{1}^{∞} f(x) dx either both converge or both diverge.

Graph: Integral Test picture (series vs area)

The rectangles represent a_n and the curve represents f(x). The integral compares “infinite area” to “infinite sum.”

10.6

p-Series (Memorize This Rule)

Σ 1/n^p converges if p>1, diverges if p≤1

Instant ID: If it looks like 1/n^p (or can be simplified into it), use p-series.

10.7

Comparison Tests (How AP expects you to argue)

Direct Comparison Test (DCT)

If 0≤a_n≤b_n and Σ b_n converges → Σ a_n converges.
If 0≤b_n≤a_n and Σ b_n diverges → Σ a_n diverges.

Limit Comparison Test (LCT) (most common on FRQ)

Choose b_n (usually p-series) and compute lim_{n→∞} a_n/b_n = c. If 0<c<∞, they behave the same (both converge or both diverge).

Score-5 move: Pick b_n by keeping only the dominant term(s). Example: (3n^2+1)/(n^4+5) behaves like 3/n^2.

10.8

Alternating Series (Leibniz Test) + Error

Alternating Series Test (AST)

For Σ (-1)^n b_n (or (-1)^{n+1}): it converges if (1) b_n decreases and (2) lim b_n = 0.

Alternating Series Error Bound: If you stop at the Nth term, then the error is at most the next term: |R_N| ≤ b_{N+1}.

Graph: Alternating partial sums “bounce” to a limit

Alternating partial sums typically overshoot/undershoot but tighten toward the limit. This is why the next-term error bound works.

10.9

Ratio & Root Tests (The factorial/exponential detectors)

Ratio Test (best when you see factorials)

L = lim |a_{n+1}/a_n|
If L<1 converge, if L>1 diverge, if L=1 inconclusive.

Root Test (best when you see n-th powers)

L = lim √[n]{|a_n|}
If L<1 converge, if L>1 diverge, if L=1 inconclusive.

Quick choice: factorials → Ratio Test. terms like (3^n) or (n^n) → Root Test.

10.10

Power Series (interval + radius of convergence)

A power series looks like:

Σ c_n (x-a)^n

What AP wants

Find the radius R and interval of convergence.
Use Ratio Test on the general term.
Always check endpoints separately (they can change!).

Graph: Convergence behavior around the center

A power series converges inside an interval centered at a: (a-R, a+R). Endpoints must be tested.

10.11

Taylor & Maclaurin Series BC

This is the BC “boss level.” But the pattern is consistent: memorize the core Maclaurin series and build everything else using algebra + substitution + differentiation/integration.

Core Maclaurin Series (must know)

Function	Series
e^x	Σ x^n/n!
sin x	Σ (-1)^n x^{2n+1}/(2n+1)!
cos x	Σ (-1)^n x^{2n}/(2n)!
1/(1-x)	Σ x^n (\|x\|<1)
ln(1+x)	Σ (-1)^{n+1} x^n/n (\|x\|<1, x>-1)
arctan x	Σ (-1)^n x^{2n+1}/(2n+1) (\|x\|≤1 endpoints special)

Typical FRQ asks: (1) write a series for a new function, (2) radius/interval of convergence, (3) approximate a value, (4) error bound using next term.

Score-5 Pattern: substitution

Want a series for sin(3x)? Replace x with 3x in the sin series.
Want a series for ∫ sin x / x dx? Use series for sin x, divide by x, integrate term-by-term.

Graph: Taylor polynomial approximating sin(x)

Taylor polynomials are accurate near the center (here: 0). AP will ask about “error” and “how many terms.”