➗Calculus II Unit 5 – Sequences and Series

Key Concepts and Definitions

• Sequences are ordered lists of numbers, denoted as $a_n$ where $n$ is a non-negative integer
• Terms of a sequence are the individual numbers in the sequence, with the $n$-th term denoted as $a_n$
• Arithmetic sequences have a constant difference $d$ between consecutive terms, where $a_{n+1} = a_n + d$
  • Example: 2, 5, 8, 11, ... is an arithmetic sequence with $d = 3$
• Geometric sequences have a constant ratio $r$ between consecutive terms, where $a_{n+1} = a_n \cdot r$
  • Example: 3, 6, 12, 24, ... is a geometric sequence with $r = 2$
• Series are the sum of the terms in a sequence, denoted as $\sum_{n=1}^{\infty} a_n$
• Partial sums are the sums of the first $n$ terms of a series, denoted as $S_n = \sum_{i=1}^{n} a_i$
• Convergence occurs when the limit of a sequence or series exists and is a finite value
• Divergence occurs when the limit of a sequence or series does not exist or is infinite

Types of Sequences

• Constant sequences have the same value for all terms, where $a_n = c$ for some constant $c$
• Recursive sequences define each term based on the previous term(s), such as the Fibonacci sequence where $a_n = a_{n-1} + a_{n-2}$
• Explicit sequences have a formula that directly calculates the $n$-th term, such as $a_n = 2n + 1$
• Monotonic sequences are either increasing ($a_{n+1} \geq a_n$) or decreasing ($a_{n+1} \leq a_n$) for all $n$
  • Example: 1, 2, 3, 4, ... is an increasing sequence
  • Example: 10, 8, 6, 4, ... is a decreasing sequence
• Bounded sequences have an upper and/or lower bound, where there exist constants $M$ and $m$ such that $m \leq a_n \leq M$ for all $n$
• Oscillating sequences alternate between increasing and decreasing or have no consistent pattern
  • Example: 1, -1, 1, -1, ... is an oscillating sequence

Convergence and Divergence

• Convergent sequences approach a specific finite value as $n$ approaches infinity
  • Example: $a_n = \frac{1}{n}$ converges to 0 as $n \to \infty$
• Divergent sequences do not approach a specific finite value or grow without bound as $n$ approaches infinity
  • Example: $a_n = n$ diverges to infinity as $n \to \infty$
• The limit of a convergent sequence is the value it approaches as $n$ approaches infinity, denoted as $\lim_{n \to \infty} a_n$
• Cauchy sequences are sequences where the terms become arbitrarily close to each other as $n$ increases
  • Formally, for any $\varepsilon > 0$, there exists an $N$ such that $|a_n - a_m| < \varepsilon$ for all $n, m > N$
• Every convergent sequence is a Cauchy sequence, but not every Cauchy sequence converges (in general metric spaces)
• Absolute convergence occurs when the series $\sum_{n=1}^{\infty} |a_n|$ converges
• Conditional convergence occurs when a series converges, but not absolutely

Series and Their Properties

• Arithmetic series are the sum of an arithmetic sequence, with the formula $S_n = \frac{n}{2}(a_1 + a_n)$
• Geometric series are the sum of a geometric sequence, with the formula $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$
  • Geometric series converge if $|r| < 1$ and diverge if $|r| \geq 1$
• Telescoping series are series where most terms cancel out, leaving only a few terms that determine the sum
  • Example: $\sum_{n=1}^{\infty} (\frac{1}{n} - \frac{1}{n+1})$ telescopes to $1$
• The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, despite the terms approaching 0
• The alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges to $\ln(2)$
• The comparison test compares a series to a known convergent or divergent series to determine its convergence or divergence
• The limit comparison test compares the limit of the ratio of corresponding terms of two series to determine convergence or divergence

Tests for Convergence

• The nth term test states that if $\lim_{n \to \infty} a_n \neq 0$, then $\sum_{n=1}^{\infty} a_n$ diverges
  • Note: The converse is not true; if the limit is 0, the series may converge or diverge
• The 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 $\int_1^{\infty} f(x) dx$ diverges, then $\sum_{n=1}^{\infty} f(n)$ diverges
• The ratio test compares 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 absolutely
  • If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| > 1$, the series diverges
• The root test compares 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 absolutely
  • If $\lim_{n \to \infty} \sqrt[n]{|a_n|} > 1$, the series diverges
• The alternating series test states that if an alternating series has terms that decrease in absolute value and approach 0, it converges
• The p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges for $p > 1$ and diverges for $p \leq 1$

Power Series

• Power series are series of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where $c$ is the center and $a_n$ are coefficients
• The interval of convergence is the range of $x$ values for which the power series converges
  • Endpoints must be checked separately using other tests
• The radius of convergence $R$ is half the length of the interval of convergence, found using the ratio test
  • $R = \lim_{n \to \infty} |\frac{a_n}{a_{n+1}}|$, if the limit exists
• Power series can be differentiated and integrated term by term within the interval of convergence
• Taylor series are power series approximations of functions centered at a specific point $c$, with coefficients $a_n = \frac{f^{(n)}(c)}{n!}$
  • Example: The Taylor series for $e^x$ centered at $c=0$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$
• Maclaurin series are Taylor series centered at $c=0$

Applications in Calculus

• Power series can be used to represent and approximate functions near a specific point
• Taylor polynomials are finite approximations of functions using Taylor series, often used for approximation and error analysis
  • The nth degree Taylor polynomial of $f(x)$ centered at $c$ is $P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!}(x-c)^k$
• The Lagrange error bound estimates the maximum error between a function and its Taylor polynomial on an interval
  • If $|f^{(n+1)}(x)| \leq M$ on $[a, b]$, then the error is bounded by $\frac{M}{(n+1)!}|x-c|^{n+1}$
• Series can be used to solve differential equations by assuming a power series solution and finding the coefficients
  • Example: The power series solution to $y' - y = 0$ with $y(0) = 1$ is $y = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
• Fourier series represent periodic functions as an infinite sum of sines and cosines, useful in signal processing and solving PDEs
  • The Fourier series of a function $f(x)$ on $[-\pi, \pi]$ is $\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$

Common Pitfalls and Tips

• Be careful with the index shift when working with sequences and series, as the first term may correspond to $n=0$ or $n=1$
• Always check the endpoints of the interval of convergence separately, as they may behave differently from the interior
• Remember that the harmonic series diverges, despite the terms approaching 0
• The comparison test and the limit comparison test are most effective when the compared series are similar in behavior
• When using the ratio test or the root test, if the limit equals 1, the test is inconclusive, and other tests should be used
• In alternating series, the error of the partial sum is bounded by the absolute value of the next term
  • $|S - S_n| \leq |a_{n+1}|$, where $S$ is the sum of the series and $S_n$ is the nth partial sum
• When solving differential equations using power series, make sure to check if the solution satisfies the initial or boundary conditions
• Remember that Taylor series are local approximations and may not be accurate far from the center point
• When working with Fourier series, be mindful of the convergence behavior at discontinuities and the Gibbs phenomenon