# Week 4 - Alternating Series

## 1. Absolutely Convergent Series

### 1.1. Theorem

• The series $\sum a_n$ converges absolutely if the series $\sum \lvert a_n \lvert$ converges.

### 1.2. Examples

• $\sum \frac{(-1)^n}{n^2}$
• $\sum \lvert \frac{(-1)^n}{n^2} \rvert = \sum \frac{1}{n^2}$
• And we have proven that $\sum \frac{1}{n^2}$ converges.
• So $\sum \frac{(-1)^n}{n^2}$ converges absolutely.

### 1.3. Procedures to Analyze Series

• Check $\lim_{n \to \infty} a_n$,
• if $\ne 0$ then diverge.
• else if $a_n \ge 0$, apply usual convergence tests, like Ratio Test, Root Test and Integral Test.
• else: Apply tests to $\sum \lvert a_n \rvert$. Try to prove the series convergence.

## 2. Conditional Convergence

### 2.1. Theorem

• $\sum a_n$ is conditional convergence,
• if $\sum a_n$ converges,
• but $\sum \lvert a_n \rvert$ diverges.

### 2.2. Examples

• $\sum \frac{(-1)^n}{n}$
• $\sum \lvert \frac{(-1)^n}{n} \rvert = \sum \frac{1}{n}$ diverges.
• But it converges, because $\sum \frac{(-1)^n}{n} = - \log 2$

## 3. Alternating Series

### 3.1. Definition

• An alternating series is a series whose terms are alternately positive and negative. Here are some examples:
• $\sum \frac{(-1)^{n+1} } {n}$ is alternating series.
• = $\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \ldots$
• $\sum \frac{(-1)^{n} \cdot \sin n } {n}$ IS NOT.
• $\sum \frac{(-1)^{n} \cdot \sin^2 n } {n}$ is.

### 3.2. Alternating Series Test

#### Theorem

• If the alternating series $\sum_{n=1}^{\infty} (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + b_5 - \cdots b_n > 0$ satisfies \begin{aligned}& b_{n+1} \le b_n \ \text{for all}\ n \\ & \lim_{n \to \infty} b_n = 0\end{aligned},
• Then, the series converges.

#### Proven

• From this figure, we can get:
• $s_{2n}$ is increasing and bounded above by $s_1$, so $\lim_{n \to \infty} s_{2n}$ exists.
• $s_{2n-1}$ is decreasing and bounded below by $s_2$, so $\lim_{n \to \infty} s_{2n-1}$ exists.
• And we know $\lim_{n \to \infty}(s_{2n} - s_{2n-1}) = \lim_{n \to \infty}(-b_{2n})$, and our conditions already says $\lim_{n \to \infty}(b_{n}) = 0$
• So, $\lim_{n \to \infty}(s_{2n}) - \lim_{n \to \infty}(s_{2n-1}) = 0$, then we can say $\lim_{n \to \infty}(s_{n})$ also exits and $= \lim_{n \to \infty}(s_{2n}) = \lim_{n \to \infty}(s_{2n-1}) = s$.
• And also, we know, alternating series, is between neighboring partial sums: $s_{2n-1} \le s \le s_{2n}$

#### Examples

• Use the fact: alternating series, is between neighboring partial sums: $s_{2n-1} \le s \le s_{2n}$ to prove $e$ is irrational.
• Definition of Irrational: A real number x is irrational if it cannot be expressed as $\frac{p}{q}$ for p, q are integers.
• Let's declare $\frac{1}{e} = \sum \frac{(-1)^n}{n!}$ (will learn it in Week 6), so our goal is, to prove $\sum \frac{(-1)^n}{n!}$ irrational.
• $\sum \frac{(-1)^n}{n!}$ is an alternating series. so $s_b < \frac{1}{e} < s_{b+1}$
• So we can get $0 < \lvert \frac{1}{e} - s_b \rvert < \frac{1}{(b+1)!}$ base on $s_{b+1} - s_b = \frac{1}{(b+1)!}$
• Then $0 < b! \cdot \lvert \frac{1}{e} - s_b \rvert < \frac{b!}{(b+1)!} = \frac{1}{b+1}$
• Assume we can rewrite $\frac{1}{e}$ to $\frac{a}{b}$: $0 < \lvert b! \cdot \frac{a}{b} - b! \cdot s_b \rvert < \frac{1}{b+1} < 1$
• We know $b! \cdot \frac{a}{b}$ is an integer. And $b! \cdot s_b = b! \sum_{n=0}^{b} \frac{(-1)^{n+1} }{n!} = \sum_{n=0}^{b} \frac{(-1)^{n+1} \cdot b!}{n!}$
• b is at least as big as n, so $\sum_{n=0}^{b} \frac{(-1)^{n+1} \cdot b!}{n!}$ is an integer too.
• So $\lvert b! \cdot \frac{a}{b} - b! \cdot s_b \rvert$ should be an integer too. But $0 < \lvert b! \cdot \frac{a}{b} - b! \cdot s_b \rvert < 1$. That's a contradiction.
• So $e$ must be irrational.

## 4. Limit Comparison Test

### 4.1. Theorem

• If $a_n \ge 0$, $b_n \ge 0$ and $\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0$,
• Then, $\sum b_n$ converges, if and only if $\sum a_n$ converges.

## 5. One of the Practice Quiz

• You may remember that $\frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}.$ By computing some terms of the (alternating!) Taylor series for arctangent, approximate $\arctan(\frac{1}{2})$ to within $\frac{1}{33}$

• From the formula for geometric series shows that $1+y+y^2+y^3+\cdots = \frac{1}{1-y}\qquad\text{if } \lvert y \rvert \lt 1.$
• And we know $x = \frac{1}{2} < 1$, so we can plug in $-x^2$ for $y$, we get that \begin{aligned} \frac{1}{1+x^2} &= \frac{1}{1-(-x^2)} \\ &= 1 + (-x^2) + (-x^2)^2 + (-x^2)^3 + \cdots + (-x^2)^n + \cdots\\ &= 1 - x^2 + x^4 - x^6 + x^8 - x^{10} + \cdots \end{aligned}
• So we have $\frac{d}{dx}\arctan(x) = 1 - x^2 + x^4 - x^6 + x^8 - x^{10}+\cdots\qquad\text{if }|x|\lt 1$
• Then \begin{aligned} \arctan(x) &= \int\left(\frac{d}{dx}\arctan (x)\right)\,dx \\ &= \int\left(1 - x^2 + x^4 - x^6 + x^8 - x^{10}+\cdots\right)\,dx\\ &= \int\left(\sum_{n=0}^{\infty}(-1)^{n}x^{2n}\right)\,dx\\ &= \sum_{n=0}^{\infty}\left(\int (-1)^{n}x^{2n}\,dx\right)\\ &= \sum_{n=0}^{\infty}\left((-1)^{n}\int x^{2n}\,dx\right)\\ &= \sum_{n=0}^{\infty}\left((-1)^{n}\frac{x^{2n+1} }{2n+1}\right) + C\\ &= C + \left( x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \frac{x^{11} }{11} +\cdots\right). \end{aligned}
• Evaluating at $x = 0$ gives $0 = \arctan(0) = C$, so we get $\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \frac{x^{11} }{11} + \cdots,\qquad\text{if }|x|\lt 1.$
• $\frac{x^5}{5} < \frac{1}{33}$, so the approximate value of $\arctan(\frac{1}{2}) = \frac{1}{2} - \frac{1}{24} = \frac{11}{24}$
• Note we are assuming $|x| < 1$, but arctangent is defined for all real numbers. The series we have here is $\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1} }{2n+1}.$

• Using the Ratio Test, we have that \begin{aligned} \lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|} &= \lim_{n\to\infty}\frac{\quad\frac{|x|^{2n+3} }{2n+3}\quad}{\frac{|x|^{2n+1} }{2n+1} }\\ &= \lim_{n\to\infty}\frac{(2n+1)|x|^{2n+3} }{(2n+3)|x|^{2n+1} }\\ &= \lim_{n\to\infty}\frac{|x|^2(2n+1)}{2n+3}\\ &= |x|^2\lim_{n\to\infty}\frac{2n+1}{2n+3}\\ &= |x|^2. \end{aligned}
• By the Ratio Test, the series converges absolutely if $|x|^2 < 1$ (that is, if $|x| < 1$) and diverges if $|x| > 1$. At $x=1$ and $x=-1$, the series is known to converge. So the radius of convergence is 1, and the equality is valid for $x \in [-1,1]$ only.
• This answer is copied from https://math.stackexchange.com/questions/29649/why-is-arctanx-x-x3-3x5-5-x7-7-dots.

## 6. Rearrangement Theorem

• Let L be a real number, $\sum a_n$ is a conditional convergent.
• Then $a_n$ can be rearranged to form $b_n$ and $\sum b_n = L$.

### 6.1. Example

• $\sum \frac{(-1)^{n+1} }{n} = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \cdots$
• Take the even terms, we get $\sum \frac{1}{2n} = \frac{1}{2} \sum \frac{1}{n}$, which diverges.
• Take the odd terms, we get $\sum \frac{1}{2n-1} > \sum \frac{1}{2n-1}$, which diverges too.
• So whichever we picked, can easily get a number L as large as we want to.