Lecture 53: Series of Constants

Chapter Content

Section 1: Series of Constant Terms
Section 2: Power series
Section 3: Fourier series

Section Overview

(1) The concept and properties of series
(2) Convergence criterion for series

Common Question Types and Typical Examples

Determining the convergence and divergence of series of constant terms

47.1 Basic Concepts of Series with Constant Terms

47.1.1 Basic Concepts

#####Definition: Constanttermseries

description: $$\sum_{n=1}^\infty u_n=u_1+u_2+\cdots+u_n+\cdots $$

Explanation

Concept:
series of constants indicates infinite terms, each of which is constant;
Because it is an infinite term, a series of constant terms is a series of constant terms with**constants with infinite terms;
Indicated by $u_{1}, u_{2},\cdots u_{n},\cdots$, $u_{n}$ by general terms;
The sum of infinite terms is obtained by taking the limit of the sum of finite terms (partial sum);
Sum of Sums:
$\sum_{i=1}^{\infty}u_{i}=u_{1}+u_{2}+\cdots+u_{n}$；
Partial and Harmonious
Represents the sum of the first n terms;
$S_{n}=u_{1}+u_{2}+\cdots+u_{n}$；

#####Definition: Convergenceofseriesofconstantterms

description: $u_{1}, u_{2},\cdots u_{n},\cdots$ represents the constant term, $S_{n}=u_{1}+u_{2}+\cdots+u_{n}$ represents the sum of the first n terms;

If:

$$\lim_{n\to+\infty}S_{n}=\sum_{n=1}^\infty u_n$$

then the current constant sequence is convergent; if there is no limit, it is divergent;

Explanation

Two questions:
Whether the convergence and divergence -> limit exists <- is a more central issue;
Value of the series -> value of the limit;

Geometric Sequence Analysis: $a+aq+aq^{2}+aq^{3}+\cdots+aq^{n-1}...$, a geometric sequence, where a is not equal to 0, find its convergence property;

When q is not equal to 1, $S_{n}=\frac{a(1-q^{n})}{1-q}$
When q is less than 1, its limit is 0 - > convergence;
When q is greater than 1, its limit does not exist at -> divergence;
If q equals 1
If q=1, then $a+a+a+\cdots$, infinite A sums up to be infinite, so it diverges;
If q=-1, then $a-a+a-a+a-a+\cdots$
Odd term -> limit is a;
Even term -> limit 0;
So there is no limit -> divergence;

47.1.2 Properties of Series

Basic Properties

Property 1: If $\sum_{n=1}^{\infty}u_n=S$, then $\sum_{n=1}^{\infty}k u_n=kS$.
Property 2:
If $\sum_{n=1}^{\infty}H_n=h$ and $\sum_{n=1}^{\infty}U_n=u$, then $\sum_{n=1}^{\infty}(H_n\pm U_n)=h\pm u$.
Two series of convergence that still converge after addition or subtraction;
Two divergent series, whose convergence is uncertain after addition or subtraction;
Two series that converge after addition or subtraction may not be convergent in their original series;
For example:
$1+1+1+1...$ and $-1-1-1-1-1...$
After adding: $(1-1)+(1-1)+(1-1)....$
Nature 3:
Remove or addfinite termsto the series; the convergence and divergence remain unchanged, but the value may change;
Property 4:
$$\sum u_n\text{ Convergence, any series after adding parentheses also converges, and the sum remains unchanged }$$
For example:
$u_{1}+u_{2}+u_{3}+u_{2}+u_{5}+u_{8}+u_{7}+u_{8}+......$
$S_{1},S_{2},S_{3},S_{4},S_{5},S_{6},S_{7},S_{8}......$
Then add parentheses: $(u_{1}+u_{2})+u_{3}+(u_{4}+u_{3})+(u_{6}+u_{7}+u_{8})+\cdots$
The parentheses are merged to form $v_{1}+v_{2}+v_{3}+v_{4}+\cdots$
Note:
1.After adding parentheses, convergence, the original series may not converge;
2. If the parentheses diverge after the extension, the original series must diverge;
Property 5: A necessary condition for series convergence
$$\sum_{n=1}^\infty u_n\text{ Convergence }\longrightarrow\lim_{n\to\infty}u_n=0$$
Note:
$u_n$ Approaches zero, and the series may not converge, so it is a necessary condition rather than a necessary condition;
For example, the harmonic series diverges: $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}+\cdots$;

47.2 Series Classification

Category

Constant-sign series
Positive-term series -> Lecture 48: Positive-term Series and Their Convergence Tests
Negative-term series: the negative of a positive-term series;
Sign-changing series:
A special case: alternating series -> Lecture 49: Alternating Series and General Series
The general case: series of arbitrary terms -> Lecture 49: Alternating Series and General Series

Question Type: ConvergenceAndDivergenceOfConstantTermSeries

PART 1: Problem-solving methods

Problem-Solving Steps

Step 1: Determine the type of series
Positive, interleaved, arbitrary terms
Step 2: Choose the method based on the number of levels
Series of positive terms -> Five types of criteria for convergence and dispersion are used for judgment;
alternating series -> a method;
Any term series -> one method;
Step 3: If a determination cannot be made, you can use definitions and properties to make a judgment
Definitions and properties apply to all types of series;

Problem-Solving Methods

Exams usually include multiple-choice questions, so it's better to use direct methods: find the correct option and prove it's correct, rather than using elimination and finding counterexamples one by one to prove other options are wrong;
How to find: For the current option, if it's wrong, go straight to the next one; If you're unsure about the correct or false option, go to the next option first—the correct one might be right at the back;

Common Conclusions

$$\sum_{n=1}^{+\infty}|b_{n|}\text{ Convergence }\rightarrow \sum_{n=1}^{+\infty}(b_{n})^{2}\text{ Convergence }$$