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Power Series

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Power Series - Complete Interactive Lesson

Part 1: Core Concepts

Power Series — Definition & Convergence

Part 1 of 7 — Introduction to Power Series

What Is a Power Series?

A power series centered at $x = c$ is:

$\boxed{\sum_{n=0}^\infty a_n (x - c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots}$

When $c = 0$ , this is a Maclaurin-type power series: $\sum a_n x^n$ .

Key Terminology

Term	Meaning
Center $c$	The point about which the series is expanded
Coefficients $a_n$	The constants multiplying each power
Radius of convergence $R$	Series converges for $
Interval of convergence

Three Convergence Possibilities

For any power series, exactly ONE is true:

Converges only at $x = c$ (radius $R = 0$ )
Converges for all $x$ (radius $R = \infty$ )
Converges for and diverges for (finite )

AP Tip: The Ratio Test is the primary tool for finding the radius of convergence.

Finding the Radius with the Ratio Test

For $\sum a_n (x-c)^n$ , apply the Ratio Test:

$L = \lim_{n \to \infty} ∣ \frac{a_{n + 1}}{a_{}}$

Finding Radius of Convergence

Convergence Analysis

Radius Computation

Summary

Power series: $\sum a_n(x-c)^n$ — an "infinite polynomial"
Radius found via Ratio Test: $R = 1 / \lim ∣ a_{n + 1} / a_{n} ∣$

Part 2: Worked Examples

Power Series — Interval of Convergence

Part 2 of 7 — Endpoint Testing

From Radius to Interval

After finding $R$ , the open interval $( c - R,\ c + R )$ is guaranteed. But endpoints need individual testing.

Endpoint Testing Procedure

Find $R$ using Ratio/Root Test

Part 3: Problem-Solving Patterns

Power Series — Operations

Part 3 of 7 — Differentiation, Integration, and Manipulation

Term-by-Term Differentiation

If $f(x) = \sum_{n=0}^\infty a_n (x-c)^n$ with radius , then:

Part 4: Graphs and Interpretation

Power Series — Function Representation

Part 4 of 7 — Building Series from Known Functions

The Essential Known Series

Memorize these — they're the building blocks:

Function	Series	IOC
$\frac{1}{1-x}$	$\sum_{n = 0}^{\infty}$

Part 5: Applications

Power Series — Differential Equations & AP Strategies

Part 5 of 7 — Series Solutions & Exam Techniques

Power Series Solutions to DEs

Some AP problems ask you to find coefficients of a power series solution to a DE.

Setup: Assume $y = \sum a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$

Part 6: Exam Strategy

Power Series — Problem-Solving Workshop

Part 6 of 7 — Mixed Practice

Workshop Focus Areas

Skill	What to Practice
Finding $R$	Ratio Test on coefficients
Endpoint testing	Substitute $x = c \pm R$ , test convergence
Series manipulation	Substitution, differentiation, integration
Coefficient extraction

Part 7: Mixed Review

Power Series — Comprehensive Review

Part 7 of 7 — Complete Topic Review

Power Series Checklist

Skill	Key Points
Definition	$\sum a_n(x-c)^n$ ; converges in interval around

L = lim_{n \to \infty} \frac{a _{n + 1}}{a _{n}} \cdot ∣ x - c ∣

Converges when $L < 1$ , i.e., $|x - c| < \frac{1}{\lim |a_{n+1}/a_n|}$ .

$\boxed{R = \frac{1}{\lim_{n\to\infty} |a_{n+1}/a_n|}}$

Example: $\sum_{n=0}^\infty \frac{x^n}{n!}$

$\left|\frac{a_{n+1}}{a_n}\right| = \frac{1}{n+1} \to 0$

So $L = 0 \cdot |x| = 0 < 1$ for all $x$ . Radius $R = \infty$ . (This is the series for $e^x$ .)

Example: $\sum_{n=0}^\infty n! x^n$

$\left|\frac{a_{n+1}}{a_n}\right| = n+1 \to \infty$

So $L = \infty$ for any $x \neq 0$ . Radius $R = 0$ . Converges only at $x = 0$ .

R = 1/ lim ∣ a_{n + 1} / a_{n} ∣

Three possibilities:

R = 0

R = \infty

, or finite

R

Endpoints must ALWAYS be tested separately

Next: Part 2 — Interval of Convergence (Endpoint Testing).

Substitute

x = c - R

into

\sum a_n(x-c)^n

→ get a numerical series

Substitute

x = c + R

→ get another numerical series

Test each for convergence (often

p

-series, alternating, geometric, etc.)

Complete Example: $\sum_{n=1}^\infty \frac{x^n}{n}$

Step 1: $|a_{n+1}/a_n| = n/(n+1) \to 1$ . So $R = 1$ .

Step 2: At $x = 1$ : $\sum 1/n$ — diverges (harmonic)

Step 3: At $x = -1$ : $\sum (-1)^n/n$ — converges (alternating harmonic)

$\boxed{\text{Interval of convergence: } [-1, 1)}$

AP Tip: The interval notation matters! Use brackets $[$ for included endpoints, parentheses $($ for excluded.

Common Endpoint Patterns

Series	$R$	At $x = c+R$	At $x = c-R$	IOC
$\sum x^n/n$	1	$\sum 1/n$ div.	$\sum(-1)^n/n$ conv.

Key Insight

At the positive endpoint ( $x = c + R$ ): all terms are positive → test with $p$ -series, comparison, etc.

At the negative endpoint ( $x = c - R$ ): signs alternate → often use AST.

Common outcome: one endpoint includes (alternating convergence), one excludes (divergence).

Endpoint Testing Practice

Interval Determination

Find the Left Endpoint

Summary

After finding $R$ , always test both endpoints
Positive endpoint often yields a positive-term series
Negative endpoint often yields an alternating series
Four possible IOC shapes: $(a,b)$ , $[a,b)$ , $(a,b]$ , $[a,b]$
AP exam ALWAYS expects endpoint testing — don't skip it!

Next: Part 3 — Operations on Power Series.

$\boxed{f'(x) = \sum_{n=1}^\infty n a_n (x-c)^{n-1}}$

The derivative has the same radius $R$ (but possibly different endpoint behavior).

Term-by-Term Integration

$\boxed{\int f(x)\,dx = C + \sum_{n=0}^\infty \frac{a_n (x-c)^{n+1}}{n+1}}$

Also has radius $R$ (but possibly different endpoint behavior).

Example: From Geometric to $\ln$

$\frac{1}{1-x} = \sum_{n=0}^\infty x^n,\quad |x| < 1$

Integrate both sides:

$-\ln(1-x) = \sum_{n=0}^\infty \frac{x^{n+1}}{n+1} = \sum_{n=1}^\infty \frac{x^n}{n}$

So $\ln(1-x) = -\sum_{n=1}^\infty \frac{x^n}{n}$ , or equivalently $\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n}$ .

AP Tip: Deriving series by differentiating/integrating known series is a VERY common AP technique.

Substitution

Replace $x$ with an expression in a known series:

$\frac{1}{1-x} = \sum x^n \implies \frac{1}{1+x^2} = \sum (-x^2)^n = \sum (-1)^n x^{2n}$

Then integrate: $\arctan x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1}$

Addition and Multiplication

Addition: $\sum a_n x^n + \sum b_n x^n = \sum (a_n + b_n) x^n$ (radius = min of the two)

Radius Under Operations

Operation	New Radius
Differentiation	Same $R$
Integration	Same $R$
Substitution $x \to g(x)$	Solve $
Addition	$\min$

Operations Practice

Manipulation Techniques

Series Derivation

Summary

Differentiate and integrate power series term by term
Radius stays the same (endpoints may change)
Substitution lets you build new series from known ones
Key chain: $1/(1-x) \to \ln(1-x) \to \arctan x$ via integration/substitution

Next: Part 4 — Representing Functions as Power Series.

\sum_{n = 0}^{\infty} x^{n}

$\boxed{\text{Most AP power series problems reduce to manipulating these six.}}$

AP Tip: You'll often need to find a series by relating the function to one of these through substitution, differentiation, or integration.

Technique: Partial Fractions + Geometric

Find the series for $f(x) = \frac{3}{2-x}$ :

$\frac{3}{2-x} = \frac{3}{2} \cdot \frac{1}{1 - x/2} = \frac{3}{2} \sum_{n=0}^\infty \left(\frac{x}{2}\right)^n = \sum_{n=0}^\infty \frac{3 x^n}{2^{n+1}}$

IOC: $|x/2| < 1 \implies |x| < 2$

Technique: Composition

Find the series for $e^{x^2}$ :

$e^u = \sum u^n/n!$ . Set $u = x^2$ :

$e^{x^2} = \sum_{n=0}^\infty \frac{x^{2n}}{n!} = 1 + x^2 + \frac{x^4}{2} + \frac{x^6}{6} + \cdots$

Technique: Integration of Known Series

Find the series for $\int_0^x e^{-t^2}\,dt$ (no elementary form!):

$e^{-t^2} = \sum (-1)^n t^{2n}/n!$ . Integrate:

This is related to the error function $\text{erf}(x)$ — series representation gives exact computation!

Function Representation

Coefficient Finding

Summary

Six essential series to memorize (geometric, $e^x$ , $\sin$ , $\cos$ , $\ln$ , $\arctan$ )
Build new series via substitution, multiplication, differentiation, integration
Partial fractions reduce rational functions to geometric-type series
Series let you "compute" functions with no elementary antiderivative

Next: Part 5 — Power Series and Differential Equations.

Then $y' = \sum_{n=1}^\infty n a_n x^{n-1} = a_1 + 2a_2 x + 3a_3 x^2 + \cdots$

Example: $y' = y$ , $y(0) = 1$

Substituting: $a_1 + 2a_2 x + 3a_3 x^2 + \cdots = a_0 + a_1 x + a_2 x^2 + \cdots$

Matching coefficients: $a_1 = a_0 = 1$ , $2a_2 = a_1 \Rightarrow a_2 = 1/2$ , $3a_3 = a_2 \Rightarrow a_3 = 1/6$

Pattern: $a_n = 1/n!$ → solution is $y = e^x$ ✓

AP Tip: These problems typically ask for the first 3 or 4 nonzero terms, not the general pattern.

Common AP FRQ Formats

Type 1: "Write the first four nonzero terms..."

Use known series + operations
Example: First 4 terms of $e^x \sin x$ → multiply truncated series

Type 2: "Find the coefficient of $x^n$ ..."

Use Taylor formula: $a_n = f^{(n)}(0)/n!$
Or manipulate known series

Type 3: "Use the series to approximate..."

Evaluate at specific $x$ , bound error
Use alternating series error bound when applicable

Type 4: "Find the interval of convergence"

Ratio test for $R$ , then test endpoints

Quick AP Checks

$\boxed{f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x-c)^n \implies a_n = \frac{f^{(n)}(c)}{n!}}$

So if you know the series, you know the derivatives at the center: $f^{(n)}(c) = n! \cdot a_n$

AP-Style Practice

Series & Derivatives

DE Series Solution

Summary

Power series can solve DEs by matching coefficients
$f^{(n)}(c) = n! \cdot a_n$ connects series coefficients to derivatives
AP FRQ: "first four nonzero terms" is the most common format
Build series from known ones rather than computing derivatives

Next: Part 6 — Problem-Solving Workshop.

a_n = f^{(n)}(c)/n!

a_{n} = f^{(n)} (c) / n!

Workshop Problems

Series Evaluation

Workshop Takeaways

Ratio Test is the go-to for finding $R$
Endpoint testing is mandatory on the AP exam
Building series from known ones is faster than computing derivatives
$f^{(n)}(c) = n! \cdot a_n$ is a powerful shortcut

Next: Part 7 — Comprehensive Review.

$\boxed{\text{Power series = the bridge between algebra and analysis on the AP exam.}}$

Comprehensive MC Review

Final Review Drill

Power Series — Complete Summary

You've mastered:

Definition & radius — Ratio Test for $R$ , three convergence scenarios
Endpoint testing — individual analysis, four IOC shapes
Operations — differentiate, integrate, substitute term-by-term
Known series — six essential Maclaurin series
Function representation — building new series from old
DE connections — coefficient matching for series solutions

Key Fact: Power series questions appear in 5+ MC questions and at least 1 FRQ on every BC exam. This is arguably the most important BC-specific topic.

Up Next: Taylor & Maclaurin Series — the general construction formula.

Multiplication by $x^k$ :

x^k \sum a_n x^n = \sum a_n x^{n+k}

(radius unchanged)

min (R_{1}, R_{2})

\int_0^x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1) n!}

Power Series

Power Series - Complete Interactive Lesson

Part 1: Core Concepts

Power Series — Definition & Convergence

What Is a Power Series?

Key Terminology

Three Convergence Possibilities

Finding the Radius with the Ratio Test

Summary

Part 2: Worked Examples

Power Series — Interval of Convergence

From Radius to Interval

Endpoint Testing Procedure

Part 3: Problem-Solving Patterns

Power Series — Operations

Term-by-Term Differentiation

Part 4: Graphs and Interpretation

Power Series — Function Representation

The Essential Known Series

Part 5: Applications

Power Series — Differential Equations & AP Strategies

Power Series Solutions to DEs

Part 6: Exam Strategy

Power Series — Problem-Solving Workshop

Workshop Focus Areas

Part 7: Mixed Review

Power Series — Comprehensive Review

Power Series Checklist

Example: ∑n=0∞xnn!\sum_{n=0}^\infty \frac{x^n}{n!}∑n=0∞​n!xn​

Example: ∑n=0∞n!xn\sum_{n=0}^\infty n! x^n∑n=0∞​n!xn

Complete Example: ∑n=1∞xnn\sum_{n=1}^\infty \frac{x^n}{n}∑n=1∞​nxn​

Common Endpoint Patterns

Key Insight

Summary

Term-by-Term Integration

Example: From Geometric to ln⁡\lnln

Substitution

Addition and Multiplication

Radius Under Operations

Summary

Technique: Partial Fractions + Geometric

Technique: Composition

Technique: Integration of Known Series

Summary

Common AP FRQ Formats

Quick AP Checks

Summary

Workshop Takeaways

Power Series — Complete Summary

Example: $\sum_{n=0}^\infty \frac{x^n}{n!}$

Example: $\sum_{n=0}^\infty n! x^n$

Complete Example: $\sum_{n=1}^\infty \frac{x^n}{n}$

Example: From Geometric to $\ln$