# UNIT 10: INFINITE SEQUENCES AND SERIES (BC ONLY)

# 1. INTRODUCTION

Imagine you are standing 2 meters away from a wall. You decide to walk toward the wall, but with a specific rule: your first step covers half the distance (1 meter). Your second step covers half of the remaining distance (0.5 meters). Your third step covers half of that (0.25 meters), and so on. Mathematically, the distance you travel is represented by the sum:
$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$$

This is an infinite process. Does the sum go to infinity? Or do you eventually reach the wall? This paradox, famously discussed by the Greek philosopher Zeno, introduces us to the world of Infinite Series. In AP Calculus BC, Unit 10 is often considered the "pinnacle" of the course. It transitions from the study of functions and values to the study of how functions themselves can be constructed from an infinite number of polynomial terms.

The study of sequences and series is the foundation of modern computing. Your calculator does not "know" what $\sin(0.5)$ is by looking at a triangle; it calculates a finite number of terms of an infinite polynomial to provide an answer accurate to ten decimal places. By the end of this chapter, you will understand how to determine if an infinite sum settles on a single value (converges) or grows without bound (diverges), and how to represent complex transcendental functions like $e^x$ or $\cos(x)$ as simple, infinite-degree polynomials.

# 2. ALL KEY CONCEPTS, TERMS, FOUNDATIONAL KNOWLEDGE, and PRINCIPLES

To master this unit, you must be fluent in the following terminology and concepts:

• Sequence: An ordered list of numbers $a_1, a_2, a_3, \dots, a_n, \dots$ often defined by a function $f(n)$.
• Series: The sum of the terms of a sequence, denoted as $\sum_{n=1}^{\infty} a_n$.
• Partial Sum ($S_n$): The sum of the first $n$ terms of a series. $S_n = a_1 + a_2 + \dots + a_n$.
• Convergence: A series converges if the limit of its partial sums exists and is finite ($\lim_{n \to \infty} S_n = S$).
• Divergence: A series diverges if the limit of its partial sums does not exist or is infinite.
• Geometric Series: A series where each term is found by multiplying the previous term by a constant ratio $r$.
• p-Series: A series of the form $\sum \frac{1}{n^p}$, where $p$ is a positive constant.
• Power Series: A series of the form $\sum c_n (x-c)^n$, which functions as a "polynomial of infinite degree."
• Radius of Convergence ($R$): The distance from the center $c$ within which the power series converges.
• Interval of Convergence: The set of all $x$-values for which a power series converges.
• Taylor Polynomial: A finite polynomial used to approximate a function near a point $x=c$.
• Maclaurin Series: A Taylor series centered at $c=0$.
• Lagrange Error Bound: A formula used to calculate the maximum possible error when using a Taylor polynomial to approximate a function.

# 3. IN-DEPTH EXPLANATION of EVERY CONCEPT and PRINCIPLE

# 3.1. Sequences and Their Limits

A sequence ${a_n}$ is a function whose domain is the set of positive integers. We are primarily interested in the limit of a sequence as $n \to \infty$.

If $\lim_{n \to \infty} a_n = L$, we say the sequence converges to $L$. If the limit is infinite or does not exist (e.g., $a_n = (-1)^n$), the sequence diverges.
Key Property: If a sequence is monotonic (always increasing or always decreasing) and bounded, it must converge.

# 3.2. Introduction to Series and the $n$-th Term Test

A series is the sum of a sequence. The most important distinction to make is: The limit of the terms is NOT the same as the sum of the series.

The $n$-th Term Test for Divergence:
If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ must diverge.
Warning: If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive. The series might converge, or it might diverge (like the harmonic series $\sum \frac{1}{n}$).

# 3.3. Integral and p-Series Tests

The Integral Test connects series to improper integrals. If $f(x)$ is positive, continuous, and decreasing for $x \ge 1$, then $\sum_{n=1}^{\infty} a_n$ and $\int_1^{\infty} f(x) dx$ either both converge or both diverge.

From this, we derive the p-Series Test:
The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$.

# 3.4. Comparison Tests

Sometimes we compare a messy series to a simpler "known" series (like a geometric or p-series).

1. Direct Comparison Test (DCT): If $0 \le a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $0 \le b_n \le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.
2. Limit Comparison Test (LCT): If $a_n > 0$ and $b_n > 0$, and $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then both series behave the same way.

# 3.5. Alternating Series and Absolute Convergence

An alternating series has terms that flip signs, e.g., $\sum (-1)^n a_n$.
Alternating Series Test (AST): $\sum (-1)^n a_n$ converges if:

1. $\lim_{n \to \infty} a_n = 0$
2. $a_{n+1} \le a_n$ (the terms are decreasing in magnitude).

Absolute vs. Conditional Convergence:

• Absolutely Convergent: $\sum |a_n|$ converges.
• Conditionally Convergent: $\sum a_n$ converges, but $\sum |a_n|$ diverges (Example: the alternating harmonic series).

# 3.6. The Ratio Test

This is the most powerful test for AP Calculus BC, especially for power series.
Let $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.

• If $L < 1$, the series converges absolutely.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

# 3.7. Power Series

A power series is $\sum_{n=0}^{\infty} c_n (x-c)^n$.
To find the Interval of Convergence:

1. Apply the Ratio Test to find the radius $R$.
2. Check the endpoints of the interval $(c-R, c+R)$ individually using other tests (p-series, AST, etc.).

# 3.8. Taylor and Maclaurin Series

A function $f(x)$ can be represented as a Taylor series centered at $x=c$:
$$f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \dots + \frac{f^{(n)}(c)}{n!}(x-c)^n + \dots$$
Or in sigma notation:
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$$

Common Maclaurin Series (MUST MEMORIZE):

# 3.9. Error Bounds

When we use a finite Taylor polynomial $P_n(x)$ to approximate $f(x)$, there is an error $R_n(x) = f(x) - P_n(x)$.

1. Alternating Series Error Bound: If the series is alternating and satisfies AST, the error $|R_n|$ is less than or equal to the first omitted term: $|R_n| \le |a_{n+1}|$.
2. Lagrange Error Bound: For any Taylor polynomial, the error is:
   $$|R_n(x)| \le \frac{M}{(n+1)!} |x-c|^{n+1}$$
   where $M$ is the maximum value of $|f^{(n+1)}(z)|$ on the interval between $c$ and $x$.

# 4. EXAMPLES

# Example 1: Limit of a Sequence

Find the limit of the sequence $a_n = \frac{3n^2 - 1}{2n^2 + 5n}$.
Solution:
$$\lim_{n \to \infty} \frac{3n^2 - 1}{2n^2 + 5n} = \lim_{n \to \infty} \frac{3 - 1/n^2}{2 + 5/n} = \frac{3}{2}$$
The sequence converges to $1.5$.

# Example 2: Sum of a Geometric Series

Find the sum of $\sum_{n=0}^{\infty} 3 \left(\frac{2}{5}\right)^n$.
Solution:
This is a geometric series with $a = 3(2/5)^0 = 3$ and $r = 2/5$. Since $|r| < 1$, it converges.
$$S = \frac{a}{1-r} = \frac{3}{1 - 2/5} = \frac{3}{3/5} = 5$$

# Example 3: Divergence Test

Does $\sum_{n=1}^{\infty} \frac{n}{n+1}$ converge?
Solution:
Check the $n$-th term limit: $\lim_{n \to \infty} \frac{n}{n+1} = 1$.
Since $1 \neq 0$, the series diverges by the $n$-th term test.

# Example 4: Repeating Decimals to Fractions

Express $0.777\dots$ as a fraction.
Solution:
This is $7(0.1) + 7(0.01) + 7(0.001) + \dots = \sum_{n=1}^{\infty} 7(0.1)^n$.
$a = 0.7$, $r = 0.1$.
$S = \frac{0.7}{1 - 0.1} = \frac{0.7}{0.9} = \frac{7}{9}$.

# Example 5: p-Series Application

Determine convergence of $\sum \frac{1}{\sqrt[3]{n^2}}$.
Solution:
This is $\sum \frac{1}{n^{2/3}}$. Since $p = 2/3$ and $2/3 \le 1$, the series diverges.

# Example 6: Telescoping Series

Find the sum of $\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right)$.
Solution:
Partial sum $S_k = (1 - 1/2) + (1/2 - 1/3) + \dots + (1/k - 1/(k+1))$.
All middle terms cancel: $S_k = 1 - \frac{1}{k+1}$.
$\lim_{k \to \infty} (1 - \frac{1}{k+1}) = 1$. The sum is $1$.

# Example 7: Integral Test

Show $\sum_{n=2}^{\infty} \frac{1}{n \ln n}$ diverges.
Solution:
Let $f(x) = \frac{1}{x \ln x}$. $\int_2^{\infty} \frac{1}{x \ln x} dx$.
Let $u = \ln x, du = 1/x dx$.
$\int \frac{1}{u} du = \ln|u| = \ln(\ln x)|_2^{\infty} = \infty$.
Since the integral diverges, the series diverges.

# Example 8: Limit Comparison Test

Does $\sum \frac{n+1}{n^2+2}$ converge?
Solution:
Compare with $\sum \frac{n}{n^2} = \sum \frac{1}{n}$ (Harmonic series, diverges).
$\lim_{n \to \infty} \frac{(n+1)/(n^2+2)}{1/n} = \lim_{n \to \infty} \frac{n^2+n}{n^2+2} = 1$.
Since $0 < 1 < \infty$, both diverge.

# Example 9: Ratio Test with Factorials

Test $\sum_{n=1}^{\infty} \frac{2^n}{n!}$ for convergence.
Solution:
$L = \lim_{n \to \infty} \left| \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} \right| = \lim_{n \to \infty} \frac{2}{n+1} = 0$.
Since $0 < 1$, the series converges absolutely.

# Example 10: Alternating Harmonic Series

Determine if $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ converges absolutely, conditionally, or diverges.
Solution:

1. AST: $\lim \frac{1}{n} = 0$ and $1/(n+1) < 1/n$. Converges.
2. Absolute: $\sum |a_n| = \sum \frac{1}{n}$ (Harmonic). Diverges.
   Result: Conditionally Convergent.

# Example 11: Alternating Series Error Bound

Approximate $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$ using 3 terms and find the error bound.
Solution:
$S_3 = 1 - \frac{1}{4} + \frac{1}{9} = \frac{36-9+4}{36} = \frac{31}{36}$.
Error $|R_3| \le |a_4| = \frac{1}{4^2} = \frac{1}{16} = 0.0625$.

# Example 12: Radius of Convergence

Find the radius of convergence for $\sum_{n=0}^{\infty} \frac{(x-3)^n}{2^n}$.
Solution:
Ratio Test: $\lim_{n \to \infty} \left| \frac{(x-3)^{n+1}}{2^{n+1}} \cdot \frac{2^n}{(x-3)^n} \right| = \left| \frac{x-3}{2} \right|$.
Set $\frac{|x-3|}{2} < 1 \implies |x-3| < 2$.
Radius $R = 2$.

# Example 13: Interval of Convergence

Find the interval for $\sum_{n=1}^{\infty} \frac{x^n}{n}$.
Solution:
Ratio Test gives $|x| < 1$, so $(-1, 1)$.
Check $x = 1$: $\sum \frac{1}{n}$ (Diverges).
Check $x = -1$: $\sum \frac{(-1)^n}{n}$ (Converges by AST).
Interval: $[-1, 1)$.

# Example 14: Power Series for $e^{-x^2}$

Find the Maclaurin series for $f(x) = e^{-x^2}$.
Solution:
Start with $e^u = \sum \frac{u^n}{n!}$. Let $u = -x^2$.
$e^{-x^2} = \sum_{n=0}^{\infty} \frac{(-x^2)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{n!} = 1 - x^2 + \frac{x^4}{2!} - \dots$

# Example 15: Integrating a Power Series

Use a power series to approximate $\int_0^1 \sin(x^2) dx$.
Solution:
$\sin(x^2) = x^2 - \frac{(x^2)^3}{3!} + \dots = x^2 - \frac{x^6}{6} + \dots$
$\int_0^1 (x^2 - \frac{x^6}{6}) dx = [\frac{x^3}{3} - \frac{x^7}{42}]_0^1 = \frac{1}{3} - \frac{1}{42} = \frac{14-1}{42} = \frac{13}{42}$.

# Example 16: Building Taylor Polynomial from Derivatives

Find the 2nd degree Taylor polynomial for $f(x) = \sqrt{x}$ centered at $c=4$.
Solution:
$f(4) = \sqrt{4} = 2$.
$f'(x) = \frac{1}{2\sqrt{x}} \implies f'(4) = \frac{1}{4}$.
$f''(x) = -\frac{1}{4x^{3/2}} \implies f''(4) = -\frac{1}{32}$.
$P_2(x) = 2 + \frac{1}{4}(x-4) + \frac{-1/32}{2!}(x-4)^2 = 2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2$.

# Example 17: Maclaurin Series for $\cos(x)$ manipulation

Find the series for $x\cos(x)$.
Solution:
$\cos(x) = \sum \frac{(-1)^n x^{2n}}{(2n)!}$.
$x\cos(x) = x \sum \frac{(-1)^n x^{2n}}{(2n)!} = \sum \frac{(-1)^n x^{2n+1}}{(2n)!} = x - \frac{x^3}{2!} + \frac{x^5}{4!} - \dots$

# Example 18: Evaluating a Finite Sum as a Function

Evaluate $\sum_{n=0}^{\infty} \frac{1}{n!}$.
Solution:
This is the Maclaurin series for $e^x$ evaluated at $x=1$.
$e^1 = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots$
The sum is $e$.

# Example 19: Differentiating Power Series

Given $f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$, find $f'(x)$.
Solution:
$f'(x) = \sum_{n=1}^{\infty} \frac{n x^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!}$.
Let $k = n-1$. $f'(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$.
Note that $f(x) = f'(x)$, which makes sense since $f(x) = e^x$.

# Example 20: Finding the $10^{th}$ Derivative

If $f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$, find $f^{(10)}(0)$.
Solution:
The general term of a Maclaurin series is $\frac{f^{(k)}(0)}{k!} x^k$.
We want $k=10$. In the given series, $2n = 10 \implies n=5$.
The term is $\frac{(-1)^5 x^{10}}{10!}$.
Equating coefficients: $\frac{f^{(10)}(0)}{10!} = \frac{-1}{10!}$.
$f^{(10)}(0) = -1$.

# Example 21: Geometric Series in Disguise

Find a power series for $f(x) = \frac{1}{1+x^2}$.
Solution:
Use $\frac{a}{1-r}$ with $a=1, r=-x^2$.
$f(x) = 1 - x^2 + x^4 - x^6 + \dots = \sum (-1)^n x^{2n}$.

# Example 22: Series for $\arctan(x)$

Derive the series for $\arctan(x)$ by integrating $\frac{1}{1+x^2}$.
Solution:
$\arctan(x) = \int (1 - x^2 + x^4 - \dots) dx = C + x - \frac{x^3}{3} + \frac{x^5}{5} - \dots$
Since $\arctan(0) = 0$, $C=0$.
$\arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1}$.

# Example 23: Convergence of $\sum \frac{\sin(n)}{n^2}$

Solution:
Use Direct Comparison. $|\frac{\sin(n)}{n^2}| \le \frac{1}{n^2}$.
Since $\sum \frac{1}{n^2}$ converges ($p=2$), $\sum |\frac{\sin(n)}{n^2}|$ converges.
The series converges absolutely.

# Example 24: Limit of a Sequence with L'Hopital

Find $\lim_{n \to \infty} \frac{\ln(n)}{n}$.
Solution:
Treat as $f(x) = \frac{\ln x}{x}$. Type $\infty/\infty$.
Apply L'Hopital: $\lim_{x \to \infty} \frac{1/x}{1} = 0$.
The sequence converges to $0$.

# Example 25: Range of $p$ for Convergence

For what values of $p$ does $\sum_{n=1}^{\infty} \frac{n}{(n^2+1)^p}$ converge?
Solution:
Compare with $\sum \frac{n}{n^{2p}} = \sum \frac{1}{n^{2p-1}}$.
Converges if $2p-1 > 1 \implies 2p > 2 \implies p > 1$.

# Example 26: Root Test (Extension)

Does $\sum \left(\frac{3n+1}{4n-1}\right)^n$ converge?
Solution:
$\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \frac{3n+1}{4n-1} = 3/4$.
Since $3/4 < 1$, it converges.

# Example 27: Finding $c_n$ for Taylor Series

Find the coefficient $c_3$ of the Taylor series for $f(x) = \ln(x)$ centered at $c=1$.
Solution:
$f'(x) = 1/x, f''(x) = -1/x^2, f'''(x) = 2/x^3$.
$f'''(1) = 2$.
$c_3 = \frac{f'''(1)}{3!} = \frac{2}{6} = \frac{1}{3}$.

# Example 28: Lagrange Error Bound for $e^{0.1}$

Approximate $e^{0.1}$ using $P_2(x)$ and bound the error.
Solution:
$P_2(0.1) = 1 + 0.1 + \frac{(0.1)^2}{2} = 1.105$.
$|R_2(0.1)| \le \frac{M}{3!} (0.1)^3$.
$f'''(x) = e^x$. On $[0, 0.1]$, $e^x$ is max at $e^{0.1}$. Since we don't know $e^{0.1}$, use $e^1 < 3$.
Error $\le \frac{3}{6} (0.001) = 0.0005$.

# Example 29: Using Taylor Series to find Limits

Evaluate $\lim_{x \to 0} \frac{\sin x - x}{x^3}$.
Solution:
$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \dots$
$\sin x - x = -\frac{x^3}{6} + \frac{x^5}{120} - \dots$
Divide by $x^3$: $-\frac{1}{6} + \frac{x^2}{120} - \dots$
As $x \to 0$, limit is $-1/6$.

# Example 30: Convergence of a Power Series at Endpoints

Interval of $\sum \frac{(x-2)^n}{n 3^n}$.
Solution:
Ratio Test $\implies |x-2| < 3 \implies -1 < x < 5$.
$x=5: \sum \frac{1}{n}$ (Diverges).
$x=-1: \sum \frac{(-1)^n}{n}$ (Converges).
Interval: $[-1, 5)$.

# Example 31: Identifying the Series for $\ln(2)$

Evaluate $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$
Solution:
This is the Maclaurin series for $\ln(1+x)$ with $x=1$.
$\ln(1+1) = \ln 2$.

# Example 32: Approximating a Value with AST Error

How many terms of $\sum \frac{(-1)^n}{n^4}$ are needed to ensure error $< 0.001$?
Solution:
$|a_{n+1}| < 0.001 \implies \frac{1}{(n+1)^4} < \frac{1}{1000}$.
$(n+1)^4 > 1000$.
$n+1 > \sqrt[4]{1000} \approx 5.6$.
$n+1 = 6 \implies n=5$.
5 terms are needed.

# Example 33: Function Definition via Series

Let $g(x) = \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$. What is $g'(x)$?
Solution:
$g(x) = \sin x - x$ (since the first term of $\sin x$ is missing and signs are flipped).
$g'(x) = \cos x - 1$.

# Example 34: Lagrange Error with a Graph

Suppose $f^{(4)}(x) \le 10$ for all $x$. Bound the error for $P_3(2)$ if $c=0$.
Solution:
$|R_3(2)| \le \frac{10}{4!} (2-0)^4 = \frac{10}{24} \cdot 16 = \frac{160}{24} = \frac{20}{3} \approx 6.66$.

# Example 35: The Geometry of Convergence

Can a series $\sum a_n$ converge if $\lim_{n \to \infty} a_n = 0.00001$?
Solution:
No. By the $n$-th term test, the limit of the terms must be exactly zero for convergence to even be possible. If the limit is $0.00001$, the sum will eventually add $0.00001$ an infinite number of times, leading to infinity.

END OF CHAPTER