♾️AP Calculus AB/BC Unit 10 – Infinite Sequences and Series (BC Only)

Key Concepts and Definitions

• Sequence defined as an ordered list of numbers, denoted as $a_n$ where $n$ is the index or position of the term
• Series defined as the sum of the terms in a sequence, denoted as $\sum_{n=1}^{\infty} a_n$
• Partial sum of a series $S_n$ represents the sum of the first $n$ terms in the series
  • Calculated using the formula $S_n = \sum_{i=1}^{n} a_i$
• Convergence of a series occurs when the limit of the partial sums exists as $n$ approaches infinity
  • If the limit exists, the series is convergent; otherwise, it is divergent
• Common series include arithmetic series (constant difference between terms) and geometric series (constant ratio between terms)
• Limit comparison test compares the behavior of a series to a known convergent or divergent series to determine convergence or divergence

Types of Sequences and Series

• Arithmetic sequences have a constant difference $d$ between consecutive terms, following the formula $a_n = a_1 + (n-1)d$
• Geometric sequences have a constant ratio $r$ between consecutive terms, following the formula $a_n = a_1r^{n-1}$
• Harmonic series is the sum of reciprocals of positive integers, defined as $\sum_{n=1}^{\infty} \frac{1}{n}$, which is divergent
• Alternating series have terms that alternate in sign $(+, -, +, -, ...)$, often written as $\sum_{n=1}^{\infty} (-1)^{n-1}a_n$
  • Alternating series test can be used to determine convergence if certain conditions are met
• $p$-series is a series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p$ is a constant
  • Converges for $p > 1$ and diverges for $p \leq 1$
• Telescoping series is a series where most terms cancel out, leaving only a finite number of terms (often the first and last terms)

Convergence and Divergence Tests

• Divergence test states that if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ diverges
• Integral test compares a series to an improper integral to determine convergence or divergence
  • If $\int_1^{\infty} f(x) dx$ converges, then $\sum_{n=1}^{\infty} f(n)$ converges; if the integral diverges, so does the series
• Comparison test compares a series to a known convergent or divergent series
  • If $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ converges; if $\sum b_n$ diverges, then $\sum a_n$ may converge or diverge
• Limit comparison test determines convergence or divergence by evaluating the limit of the ratio of two series
• Ratio test evaluates the limit of the ratio of consecutive terms to determine convergence or divergence
  • If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| < 1$, the series converges; if the limit is $> 1$, the series diverges; if the limit is 1, the test is inconclusive
• Root test evaluates the limit of the nth root of the absolute value of the nth term to determine convergence or divergence
  • If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, the series converges; if the limit is $> 1$, the series diverges; if the limit is 1, the test is inconclusive

Working with Infinite Series

• Manipulating series involves performing operations such as addition, subtraction, multiplication, and division on series
• Cauchy product is used to multiply two series, defined as $(\sum_{n=0}^{\infty} a_n)(\sum_{n=0}^{\infty} b_n) = \sum_{n=0}^{\infty} c_n$, where $c_n = \sum_{k=0}^{n} a_kb_{n-k}$
• Differentiation and integration of series can be performed term by term, provided the resulting series converges
  • $\frac{d}{dx} \sum_{n=0}^{\infty} a_nx^n = \sum_{n=1}^{\infty} na_nx^{n-1}$ and $\int \sum_{n=0}^{\infty} a_nx^n dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}x^{n+1} + C$
• Substitution can be used to evaluate series by replacing the variable with a specific value
• Partial fractions decomposition can be used to split a rational function into simpler terms, making it easier to find the series representation

Power Series and Taylor Series

• Power series is a series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where $c$ is the center of the series
• Radius of convergence $R$ determines the interval of convergence for a power series
  • Series converges absolutely for $|x-c| < R$, converges conditionally for $|x-c| = R$, and diverges for $|x-c| > R$
• Interval of convergence is the set of $x$ values for which the power series converges
• Taylor series is a power series representation of a function centered at a specific point $c$, given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$
  • Maclaurin series is a special case of Taylor series centered at $c=0$
• Common Taylor series include expansions for $e^x$, $\sin(x)$, $\cos(x)$, and $\ln(1+x)$
• Lagrange error bound provides an upper bound for the error in a Taylor polynomial approximation

Applications in Calculus

• Series can be used to approximate functions, especially when the function is difficult to evaluate directly
• Taylor polynomials are used to approximate functions near a specific point
  • Higher-degree Taylor polynomials generally provide better approximations
• Series can be used to solve differential equations by assuming a power series solution and finding the coefficients
• Fourier series represent periodic functions as a sum of sines and cosines, useful in analyzing waveforms and signals
• Series are used in numerical integration techniques, such as the trapezoidal rule and Simpson's rule, to approximate definite integrals

Common Mistakes and Tips

• Be careful when applying convergence tests, as some tests may be inconclusive for certain series
• Remember that absolute convergence implies convergence, but conditional convergence does not imply absolute convergence
• When working with alternating series, check if the conditions for the alternating series test are satisfied before applying the test
• Pay attention to the interval of convergence when working with power series, as the series may behave differently outside this interval
• When using Taylor series approximations, consider the degree of the polynomial and the proximity to the center point to ensure accurate results
• Double-check the indices and limits when manipulating series, as incorrect indices can lead to errors in the final result
• Practice various types of series problems to develop a strong understanding of the concepts and techniques involved

Practice Problems and Solutions

1. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{n}{n^2+1}$.

   • Solution: The series converges by the limit comparison test with $\sum_{n=1}^{\infty} \frac{1}{n}$.

2. Find the interval of convergence for the power series $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n^2}$.

   • Solution: The radius of convergence is $R=1$, and the interval of convergence is $1 \leq x \leq 3$.

3. Use the ratio test to determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{n!}{n^n}$.

   • Solution: The series converges by the ratio test, as $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = \lim_{n \to \infty} \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} = \lim_{n \to \infty} \frac{n^n}{(n+1)^n} = \frac{1}{e} < 1$.

4. Find the Taylor series for $f(x)=\cos(x)$ centered at $c=0$, and determine the interval of convergence.

   • Solution: The Taylor series for $\cos(x)$ centered at $c=0$ is $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}$, and the interval of convergence is $(-\infty, \infty)$.

5. Evaluate the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$ using the alternating series test.

   • Solution: The series converges by the alternating series test, as $\lim_{n \to \infty} \frac{1}{n} = 0$ and $\frac{1}{n}$ is decreasing.