Lecture 55: Alternating Series

49.1 alternating series

49.1.1 Concept of alternating series

#####Definition: Alternatingseries

description: $$\sum_{n=1}^\infty(-1)^{n-1}u_n,u_n>0$$

Explanation

• Series with alternating positive and negative occurrences:
• 1. $u_{1}-u_{2}+u_{3}-u_{4}+u_{5}-u_{6}+\cdots$；
• 2. $-u_{1}+u_{2}-u_{3}+u_{4}-u_{5}+u_{6}\cdots$；
• This series is called an alternating series;
• And $u_n >=0$

Note

• If the number of terms is infinite,parentheses cannot be arbitrarily added, andorder cannot be arbitrarily changed**;
• For example:
• $1-1+1-1+1-1+1-1+1-1+1-1+1......$
• If it is an odd term ->, the result is 1;
• If the even term -> is 0;
• If you add parentheses
• $(1-1)+(1-1)+(1-1)+(1-1)+......$
• Adding parentheses to the infinite term like this may cause problems because it is unclear whether it is an odd or even term;

49.1.2 Methods for Determining alternating series

#####Theorem: Leibnizprinciple

Description: currently has an alternating series $\sum_{n=1}^{+\infty}\left(-1\right)^{n-1}u_{n}$, when:

1) $u_{n}\geq u_{n+1}$: Monotonous subtraction

2）$\lim_{n\to\infty}u_{n}=0$

then $\sum_{n=1}^{+\infty}\left(-1\right)^{n-1}u_{n}$ series converges, S <= $u_1$, and $|r_{n}|\leq u_{n+1}$

Explanation

• Note:
• Requirements:
• 1) $u_{n}\geq u_{n+1}$: Monotonous subtraction
• 2）$\lim_{n\to\infty}u_{n}=0$
• These two requirements aresufficient conditionsfor the convergence of an alternate series, meaning if either of these does not hold, the convergence of an alternate series is also possible;
• Premise:
• Before using Leibniz's theorem, it must be ensured that the current is an alternating series;
• The latter is smaller than the former;
• And $u_n$ approaches zero;

Example: The Leibniz criterion is a sufficient condition

• $$\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2^{n+(-1)^n}}\text{ Convergence, but }u_n=\frac1{2^{n+(-1)^n}}\text{ and did not decrease in number }.$$

49.2 Arbitrary Term Series

49.2.1 Concept of Arbitrary Term Series

Concept of Arbitrary Term Series

• Features:
• There are positive terms and negative terms;
• And both positive and negative terms must be infinite;
• Any term: $u_{1}+u_{2}+u_{3}+u_{4}+\cdots$, and the positive or negative of $u_n$ is unknown;
• Positive term series: $|u_{1}|+|u_{2}|+|u_{3}|+|u_{4}|+\cdots$, each term takes an absolute value;
• This is an absolute series;

49.2.2 Method for Determining Arbitrary Term Series

Principle: Convert new problems into old problems -> Convert a variant series into a positive term series -> Add an absolute value to the general term -> When this new positive term series converges, any term series also converges;

#####Definition: Absoluteconvergenceandconditionalconvergence

description:

1. Absolute convergence: If the absolute value of $\sum|u_{n}|$ converges, it is called $\sum u_n$ convergence;

2. Conditional convergence: If $u_n$ is convergent and $\sum|u_{n}|$ divergent, then $\sum u_n$ is called conditionally convergent;

Details:

(1) If $\sum^{\infty}_{n=1}=|a_n|$ converges, then $\sum^{\infty}_{n=1}=a_n$ must converge; in this case, $\sum^{\infty}_{n=1}a_n$ is called absolute convergence

(2) If $\sum^{\infty}_{n=1}=a_n$ converges and $\sum^{\infty}_{n=1}=|a_n|$ diverges, then it is called conditional convergence $\sum^{\infty}_{n=1}a_n$

Explanation

• Absolute convergence
• $\sum|u_{n}|$ Convergence
• $\sum u_n$ Convergence
• Conditional convergence
• $\sum|u_{n}|$ divergence
• $\sum u_n$ Convergence

Examples

• $\sum(-1)^{n-1}\frac{1}{n}$: Interleaved adjustment series -> convergent;
• $\sum(-1)^{n}\frac{1}{n^{3}}$: It is absolutely convergent;

Key Conclusions

• 1. A series of absolute convergence must converge -> that is: $\sum^{\infty}_{n=1}|a_n|$ converges -> $\sum^{\infty}_{n=1}a_n$ converges;
• 2. A series formed by all positive (or negative) terms of conditionally convergent series must diverge -> $$\sum_{n=1}^\infty u_n\text{ Conditional convergence }\Rightarrow\sum_{n=1}^\infty\frac{u_n+|u_n|}2\text{ and }\sum_{n=1}^\infty\frac{u_n-|u_n|}2\text{ Diverges }.$$

#####Theorem: Absoluteconvergenceofanytermseries

Description: if each term is summed with absolute value $\sum_{n=1}^{+\infty}|u_{n}|$, it converges, then its original series $\sum_{n=1}^{+\infty}u_{n}$ is also convergent;

Explanation

• Simpler understanding:
• If all numbers are positive (plus absolute value), it should converge even more;

#####Theorem: Judgmentfordivergenceandconvergenceofanytermseries

Description: if $\sum_{n=1}^{+\infty}u_{n}=u_{1}+u_{2}+u_{3}+\cdots$ is an arbitrary series and $\lim_{n\to+\infty}\left|\frac{u_{n+1}}{u_{n}}\right|=l$

1) When $l$ < 1, $\sum u_n$ converges absolutely;

2) When $L>1(+\infty)$, $\sum u_n$ diverges;

3) When L = 1, it cannot be determined;

Explanation

• If L>1, then the original series is divergent;