Lernzettel: Infinite Sequences and Series

📋 Course Outline

1. Sequence Limits and Behavior
2. Infinite Series and Geometric Sums
3. Integral Test and P-Series
4. Direct and Limit Comparison
5. Alternating and Absolute Convergence
6. Choosing a Convergence Test
7. Power Series Representations
8. Taylor and Maclaurin Series
9. Standard Maclaurin Series
10. Taylor Approximation and Error

📖 1. Sequence Limits and Behavior

🔑 Key Concepts & Definitions

• Infinite sequence : An ordered list of numbers, commonly written as $\{a_n\}$, where $a_n$ denotes the nth term and the domain consists of specified positive integers.
• Sequence limit : The statement $a_n\to L$ as $n\to\infty$ means that the terms become arbitrarily close to $L$ for sufficiently large $n$; a finite limit implies convergence.

📝 Essential Points

• The sequence $\{r^n\}$ converges precisely when $-1<r\leq1$, with limit $0$ for $-1<r<1$ and limit $1$ for $r=1$.

• The Monotonic Sequence Theorem states that every bounded monotonic sequence converges, specifically an increasing sequence bounded above or a decreasing sequence bounded below.

📖 2. Infinite Series and Geometric Sums

🔑 Key Concepts & Definitions

• Infinite series : $\sum_{n=1}^{\infty}a_n$ is defined through its partial sums $s_n=\sum_{i=1}^{n}a_i$, and it converges when $s_n$ approaches a finite real number.

★ Must-know

📐 Formula — A geometric series satisfies $\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{1-r}$ when $|r|<1.$

📌 If $\lim_{n\to\infty}a_n$ does not exist or is not zero, then $\sum a_n$ diverges by the Test for Divergence.

Further detail

• The harmonic series $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges even though its terms approach zero.

💡 Memory Hook

Terms → partial sums → total.

📖 3. Integral Test and P-Series

★ Must-know

📌 The Integral Test applies when $a_n=f(n)$ and $f$ is continuous, positive, and decreasing on the relevant tail; then $\sum a_n$ and $\int_1^{\infty}f(x)\,dx$ either both converge or both diverge.

📐 Formula — The p-series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ converges when $p>1$ and diverges when $p\leq1.$

📐 Formula — For a convergent positive decreasing series with $a_n=f(n)$, the remainder satisfies $\int_{n+1}^{\infty}f(x)\,dx\leq R_n\leq\int_n^{\infty}f(x)\,dx.$

Further detail

• For $\sum 1/n^3$, the bound $R_n\leq1/(2n^2)$ requires at least 32 terms to guarantee an error below $0.0005$.

📖 4. Direct and Limit Comparison

★ Must-know

📌 For positive terms, Direct Comparison proves convergence when $0\leq a_n\leq b_n$ and $\sum b_n$ converges, and proves divergence when $a_n\geq b_n\geq0$ and $\sum b_n$ diverges.

📌 Limit Comparison applies to positive-term series when $\lim_{n\to\infty}\frac{a_n}{b_n}=c$ with $0<c<\infty$; the two series then have the same convergence behavior.

Further detail

• For rational or algebraic terms, compare dominant powers of $n$ with a known p-series, using Direct or Limit Comparison.

📌 If $0\leq a_k\leq b_k$ for all sufficiently large $k$, then the remainder of the $a_k$ series is no larger than the corresponding remainder of the $b_k$ series.

📖 5. Alternating and Absolute Convergence

★ Must-know

📌 The Alternating Series Test proves convergence of $\sum(-1)^nb_n$ when $b_n$ decreases and $b_n\to0$.

📐 Formula — For an alternating series satisfying the test conditions, the remainder obeys $|R_n|\leq b_{n+1},$ so the error is at most the first omitted magnitude.

• A series is absolutely convergent when $\sum|a_n|$ converges, and conditionally convergent when $\sum a_n$ converges but $\sum|a_n|$ diverges.

Further detail

• Absolute convergence implies ordinary convergence, and every rearrangement of an absolutely convergent series has the same sum.

💡 Memory Hook

Decrease → zero → alternating convergence.

📖 6. Choosing a Convergence Test

★ Must-know

• 🔄 Process :
  1. To select a convergence test, first inspect the term limit
  2. check for a p-series or geometric form
  3. inspect signs and algebraic structure

Further detail

• The supplied material suggests the Ratio Test for factorials, products, or constant-to-the-n terms, and the Root Test for terms of the form $(b_n)^n$.

• The supplied lecture warns that the Ratio Test is not useful for many rational or p-series terms because $a_{n+1}/a_n$ tends to $1$.

💡 Memory Hook

Term limit → exact form → signs → algebra.

📖 7. Power Series Representations

★ Must-know

📐 Formula — For $|x|<1$, the geometric identity is $\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n.$

• Inside the radius of convergence, a power series may be differentiated or integrated term by term, and the resulting series has the same radius.

Further detail

• Substituting $-x^2$ into the geometric series gives $\frac{1}{1+x^2}=\sum_{n=0}^{\infty}(-1)^nx^{2n}$ for $|x|<1$.

• The supplied Bessel-function example has a power series converging for every real $x$, so its domain is all real numbers and it can be differentiated term by term.

💡 Memory Hook

Geometric template → substitution or calculus → new function series.

📖 8. Taylor and Maclaurin Series

🔑 Key Concepts & Definitions

• Maclaurin series : A Taylor series centered at $a=0$, written as $\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.$

📝 Essential Points

📐 Formula — For a Taylor series centered at $a$, the coefficient is $c_n=\frac{f^{(n)}(a)}{n!},$ and the series is $\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n.$

• The nth-degree Taylor polynomial $T_n(x)=\sum_{i=0}^{n}\frac{f^{(i)}(a)}{i!}(x-a)^i$ is a finite partial sum and need not equal the function.

📌 A function equals its Taylor series on an interval only when the remainder $R_n(x)=f(x)-T_n(x)$ approaches zero there.

💡 Memory Hook

Derivatives → coefficients → series → remainder.

📖 9. Standard Maclaurin Series

★ Must-know

📐 Formula — The Maclaurin series for the exponential function is $e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$ with radius $R=\infty$.

📐 Formula — The binomial series is $(1+x)^k=\sum_{n=0}^{\infty}\binom{k}{n}x^n$ for $|x|<1$, with $\binom{k}{n}=\frac{k(k-1)\cdots(k-n+1)}{n!}.$

Further detail

• The standard series table gives radius $\infty$ for $\sin x$ and $\cos x$, and radius $1$ for $\tan^{-1}x$ and $\ln(1+x)$.

• Known series can generate new ones by substitution, multiplication, term-by-term differentiation or integration, formal multiplication, and power-series long division.

📖 10. Taylor Approximation and Error

★ Must-know

📐 Formula — The linear Taylor approximation is $T_1(x)=f(a)+f'(a)(x-a).$

📐 Formula — Taylor’s Inequality states that if $|f^{(n+1)}(x)|\leq M$ on the relevant interval, then $|R_n(x)|\leq\frac{M|x-a|^{n+1}}{(n+1)!}.$

Further detail

• For $f(x)=x^{1/3}$ centered at $a=8$, the second-degree Taylor polynomial is $T_2(x)=2+\frac{1}{12}(x-8)-\frac{1}{288}(x-8)^2.$

• On $7\leq x\leq9$, using $M=0.0021$ for $|f'''(x)|$ gives the bound $|R_2(x)|<0.0004$ for the cube-root approximation.

• In the relativity application, expanding the relativistic kinetic-energy expression for $v\ll c$ gives the classical approximation $K\approx\frac12m_0v^2.$

💡 Memory Hook

Center → derivatives → polynomial → error bound.

📊 Synthesis Tables

Convergence Test Selection

Series form	Suggested test	Key condition
p-series	p-Series Test	p greater than 1 converges; p less than or equal to 1 diverges
Geometric	Geometric Series Test	Absolute value of r less than 1
Alternating signs	Alternating Series Test	Magnitudes decrease to zero
Positive rational or algebraic terms	Comparison Test	Compare with a known benchmark
Factorials or products	Ratio Test	Suggested by the supplied lecture
Whole nth powers	Root Test	Suggested when terms look like (b_n)^n