Power Series

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

Three Convergence Possibilities

For any power series, exactly ONE is true:

1. Converges only at $x = c$ (radius $R = 0$)
2. Converges for all $x$ (radius $R = \infty$)
3. Converges for $|x - c| < R$ and diverges for $|x - c| > R$ (finite $R > 0$)

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:

$L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \cdot |x - c|$

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 $x = 0$.

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: $R = 1/\lim|a_{n+1}/a_n|$
• Three possibilities: $R = 0$, $R = \infty$, or finite $R$
• Endpoints must ALWAYS be tested separately

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

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

1. Find $R$ using Ratio/Root Test
2. Substitute $x = c - R$ into $\sum a_n(x-c)^n$ → get a numerical series
3. Substitute $x = c + R$ → get another numerical series
4. 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

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.

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 $R > 0$, then:

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

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)
• Multiplication by $x^k$: $x^k \sum a_n x^n = \sum a_n x^{n+k}$ (radius unchanged)

Radius Under Operations

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.

Power Series — Function Representation

Part 4 of 7 — Building Series from Known Functions

The Essential Known Series

Memorize these — they're the building blocks:

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

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

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

Function Representation

Series Building

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.

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

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.

Power Series — Problem-Solving Workshop

Part 6 of 7 — Mixed Practice

Workshop Focus Areas

Workshop Problems

IOC Workshop

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.

Power Series — Comprehensive Review

Part 7 of 7 — Complete Topic Review

Power Series Checklist

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

Comprehensive MC Review

Final Review Drill

Final Challenge

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.