Lernzettel: Infinite Series and Taylor Methods

📋 Course Outline

1. Sequence Limits and Convergence
2. Geometric and Infinite Series
3. Integral and P-Series Tests
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 Convergence

🔑 Key Concepts & Definitions

• Infinite sequence : An ordered list of numbers, written as a_n, and can be viewed as a function whose domain is the positive integers or another specified integer index set.
• Sequence convergence : A sequence converges to L when its terms become arbitrarily close to L as n approaches infinity; if no finite limit exists, the sequence diverges.
• Monotonic Sequence Theorem : Every bounded monotonic sequence converges; in particular, an increasing sequence bounded above or a decreasing sequence bounded below converges.

📝 Essential Points

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

📖 2. Geometric and Infinite Series

🔑 Key Concepts & Definitions

• Infinite series : Defined through its partial sums, and it converges when the sequence of partial sums approaches a finite real number.

★ Must-know

📐 Formula — The nth partial sum is $s_n = \sum_{i=1}^{n} a_i$.

📐 Formula — The geometric series $\sum_{n=1}^{\infty} ar^{n-1}$ converges when $|r|<1$ and then has sum $\frac{a}{1-r}$.

📌 If the limit of a_n is nonzero or does not exist, then the series sum a_n diverges; if a_n approaches zero, the test is inconclusive.

Further detail

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

📖 3. Integral and P-Series Tests

★ Must-know

📌 If f is continuous, positive, and decreasing on the relevant tail and a_n=f(n), then sum a_n and the improper integral of f have the same convergence behavior.

📌 The p-series $\sum_{n=1}^{\infty}\frac{1}{n^p}$ converges when p>1 and diverges when p<=1.

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

Further detail

• For the series sum 1/n^3, the integral remainder bound gives R_n <= 1/(2n^2), so at least 32 terms guarantee an error below 0.0005.

📖 4. Direct and Limit Comparison

★ Must-know

📌 For nonnegative terms, if 0 <= a_n <= b_n and sum b_n converges, then sum a_n converges; if a_n >= b_n >= 0 and sum b_n diverges, then sum a_n diverges.

📌 For positive-term series, if the limit of a_n/b_n is c with 0<c<infinity, then sum a_n and sum b_n either both converge or both diverge.

Further detail

• For rational or algebraic terms, compare dominant powers of n with a known p-series, choosing an upper bound for convergence or a lower bound for divergence.

📌 If 0 <= a_k <= b_k for every k>n, then the tail remainder R_n of sum a_k is at most the tail T_n of sum b_k.

📖 5. Alternating and Absolute Convergence

★ Must-know

📌 The alternating series sum (-1)^n b_n converges when b_n is decreasing and approaches zero.

📐 Formula — For an alternating series satisfying the test conditions, the remainder obeys $|R_n|\le b_{n+1}$.

• 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

• Every rearrangement of an absolutely convergent series has the same sum, while rearranging a conditionally convergent series can change its sum.

💡 Memory Hook

Decrease, approach zero, then estimate the remainder.

📖 6. Choosing a Convergence Test

★ Must-know

• A series should be classified by its form:
  • first inspect the term limit
  • then check p-series or geometric forms
  • sign patterns
  • algebraic structure
  • finally factorial
  • exponential
  • integral
  • ratio
  • root patterns

Further detail

• The supplied material suggests the Ratio Test for factorials, products, or constant-to-n terms and the Root Test for terms of the form (b_n)^n, but does not provide their full theorem statements.

• The Ratio Test is warned against for p-series and rational or algebraic terms because the ratio a_(n+1)/a_n tends to 1 there.

📖 7. Power Series Representations

★ Must-know

📐 Formula — The geometric template is $\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$ for |x|<1.

• New power-series representations can be obtained by substituting an expression for x into a known series and transforming the convergence condition accordingly.

📐 Formula — Inside its radius of convergence, a power series can be differentiated term by term as $\frac{d}{dx}\sum_{n=0}^{\infty}c_n(x-a)^n=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$.

Further detail

• Term-by-term differentiation and integration preserve the radius of convergence of a power series, although endpoint behavior may require separate analysis.

📖 8. Taylor and Maclaurin Series

📝 Essential Points

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

📐 Formula — The nth Taylor polynomial centered at a is $T_n(x)=\sum_{i=0}^{n}\frac{f^{(i)}(a)}{i!}(x-a)^i$.

• A formal Taylor series equals the function only when the remainder R_n(x)=f(x)-T_n(x) approaches zero on the interval under consideration.

📐 Formula — Taylor's Inequality states that if |f^(n+1)(x)| <= M on |x-a| <= d, then $|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}$.

📖 9. Standard Maclaurin Series

★ Must-know

📐 Formula — The Maclaurin series for e^x is $e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$ and has radius of convergence infinity.

📐 Formula — The standard series include $\sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ and $\cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}$, both with infinite radius.

Further detail

📐 Formula — The standard series include $\arctan x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1}$ and $\ln(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}x^n}{n}$, each with radius 1.

📐 Formula — The binomial series is $(1+x)^k=\sum_{n=0}^{\infty}\binom{k}{n}x^n$ for |x|<1, and it terminates for positive integer k.

📖 10. Taylor Approximation and Error

★ Must-know

• To approximate with a Taylor polynomial, choose a center a, compute derivatives through order n, construct T_n, and bound the next derivative over the whole interval.

📌 Taylor approximation error can be estimated by evaluating the remainder, using the alternating-series estimate when applicable, or applying Taylor's Inequality when a derivative bound is available.

Further detail

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

• For f(x)=x^(1/3) centered at 8, the second-degree Taylor polynomial is $T_2(x)=2+\frac{1}{12}(x-8)-\frac{1}{288}(x-8)^2$, and on 7<=x<=9 the lecture obtains |R_2(x)|<0.0004.

• Using the binomial series for (1+x)^(-1/2) with x=-v^2/c^2, relativistic kinetic energy reduces for v much less than c to approximately (1/2)m_0v^2.

📊 Synthesis Tables

Convergence Test Selection

Series form	Suggested test	Key condition
Nonzero or nonexistent term limit	Test for Divergence	lim a_n is not 0
1/n^p	P-Series	p>1 converges; p<=1 diverges
a r^n	Geometric Series	|r|<1
Alternating (-1)^n b_n	Alternating Series Test	b_n decreases and tends to 0
Rational or algebraic terms	Comparison	Use dominant powers of n
Factorials or products	Ratio Test	Only source-limited guidance provided