Hoja de repaso: Infinite Series and Taylor Methods

📋 Course Outline

1. Sequences and Their Limits
2. Monotone and Bounded Sequences
3. Infinite Series and Geometric Sums
4. Integral and P-Series Tests
5. Comparison Tests and Error Bounds
6. Alternating and Absolute Convergence
7. Choosing a Convergence Test
8. Power Series and Their Calculus
9. Taylor and Maclaurin Series
10. Taylor Approximation and Applications

📖 1. Sequences and Their Limits

🔑 Key Concepts & Definitions

• Sequence : An infinite sequence is an ordered list of numbers whose nth term is commonly written as a_n, and it can be viewed as a function with integer inputs.
• Sequence Limit : A sequence a_n converges to L when its terms become arbitrarily close to L as n becomes sufficiently large; if no finite limit exists, the sequence diverges.

★ Must-know

• For r>0, the sequence 1/n^r approaches 0 as n approaches infinity.

• The sequence r^n converges precisely when -1<r<=1; its limit is 0 for -1<r<1 and 1 for r=1.

Further detail

📌 If a_n approaches L and f is continuous at L, then f(a_n) approaches f(L).

📖 2. Monotone and Bounded Sequences

🔑 Key Concepts & Definitions

• Monotonic Sequence : A sequence is monotonic when it is either increasing, with a_n<a_(n+1), or decreasing, with a_n>a_(n+1).
• Bounded Sequence : A sequence is bounded when it has both an upper bound and a lower bound.

★ Must-know

• Every bounded monotonic sequence converges; specifically, an increasing sequence bounded above and a decreasing sequence bounded below both converge.

Further detail

• The sequence a_n=n/(n+1)=1/(1+1/n) converges to 1 because 1/n approaches 0.

💡 Memory Hook

Monotonicity plus boundedness leads to convergence.

📖 3. Infinite Series and Geometric Sums

🔑 Key Concepts & Definitions

• Infinite Series : The formal sum of the terms of a sequence, and it converges when its sequence of partial sums approaches a finite number.
• Partial Sum : The nth partial sum of a series is s_n=a_1+a_2+...+a_n.

★ Must-know

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

📌 If the limit of a_n does not exist or is not zero, then the series sum a_n diverges.

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

Further detail

📐 Formula — For a geometric series with first term a and ratio r, the nth partial sum is $s_n=a\frac{1-r^n}{1-r}$ when r is not 1.

💡 Memory Hook

Terms → partial sums → series convergence.

📖 4. 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, the remainder satisfies $\int_{n+1}^{\infty}f(x)\,dx\le R_n\le\int_n^{\infty}f(x)\,dx$.

Further detail

• The series sum 1/n^3 converges, while the series sum 1/n^(1/3) diverges by the p-series rule.

📖 5. Comparison Tests and Error Bounds

★ Must-know

• For nonnegative terms, a_n<=b_n with sum b_n convergent proves sum a_n convergent, while a_n>=b_n with sum b_n divergent proves sum a_n divergent.

📌 For positive-term series, if $\lim_{n\to\infty}\frac{a_n}{b_n}=c$ with 0<c<infinity, then sum a_n and sum b_n either both converge or both diverge.

Further detail

• For rational expressions in n, comparison commonly uses the dominant powers of n to select a known p-series.

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

📖 6. Alternating and Absolute Convergence

🔑 Key Concepts & Definitions

• Absolute Convergence : A series is absolutely convergent when the series of absolute values, sum |a_n|, converges.
• Conditional Convergence : A series is conditionally convergent when sum a_n converges but sum |a_n| diverges.

★ Must-know

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

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

Further detail

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

📖 7. Choosing a Convergence Test

★ Must-know

• A convergence strategy begins by checking the term limit, then identifying an exact p-series or geometric form, inspecting signs, and finally examining algebraic structure.

Further detail

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

• The series with terms 2^k/k! is a factorial-containing form for which the lecture suggests the Ratio Test.

💡 Memory Hook

Term limit → exact form → signs → algebraic structure.

📖 8. Power Series and Their Calculus

★ Must-know

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

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

📐 Formula — Inside the radius of convergence, if $f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n$, then $f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$ and term-by-term integration is also valid.

Further detail

• Term-by-term differentiation and integration preserve the radius of convergence of a power series.

• Replacing x by -x^2 in the geometric series gives 1/(1+x^2)=sum (-1)^n x^(2n), with interval of convergence (-1,1) and radius 1.

📖 9. Taylor and Maclaurin Series

🔑 Key Concepts & Definitions

• Taylor Polynomial : The nth-degree Taylor polynomial is the finite sum $T_n(x)=\sum_{i=0}^{n}\frac{f^{(i)}(a)}{i!}(x-a)^i$.

★ Must-know

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

📐 Formula — The Taylor series of f centered at a is $\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$, and the Maclaurin series is the special case a=0.

• The Taylor remainder is $R_n(x)=f(x)-T_n(x)$, and proving that R_n(x) approaches zero establishes that the Taylor series represents f(x).

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

Further detail

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

📖 10. Taylor Approximation and Applications

🔑 Key Concepts & Definitions

• Taylor Approximation : Near its center a, a function can be approximated by its Taylor polynomial T_n(x), with T_1(x)=f(a)+f'(a)(x-a) as the linearization.

★ Must-know

• To construct and bound a Taylor approximation, choose the center, compute derivatives through order n, form T_n, bound the next derivative on the whole interval, and apply Taylor's Inequality.

Further detail

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

• For 7<=x<=9, the cube-root example uses M=0.0021 and gives |R_2(x)|<0.0004 through Taylor's Inequality.

• Using the binomial series for (1+x)^(-1/2) with x=-v^2/c^2 gives the low-speed approximation K approximately (1/2)m_0v^2 for relativistic kinetic energy.

📊 Synthesis Tables

Core Convergence Tests

Test	Required condition	Conclusion
Nth-term divergence	lim a_n is nonzero or does not exist	The series diverges
Geometric	Common ratio satisfies |r|<1	The series converges
p-series	Series is sum 1/n^p	Converges for p>1; diverges for p<=1
Alternating Series	Magnitudes decrease and approach zero	The series converges

Convergence Classification

Class	Series condition	Rearrangement behavior
Absolutely convergent	sum |a_n| converges	Every rearrangement has the same sum
Conditionally convergent	sum a_n converges but sum |a_n| diverges	Rearrangement may change the sum
Divergent	sum a_n does not converge	No finite sum is assigned