Sequences

MAT1001 (Calculus I) discussed two key topics in the analysis and computation of functions:

• derivative of a function,

• integral of a function.

We are going to talk about a third key topic: infinite series.

Infinite series can be used to

• express numbers $${\pi\over 4} = 1-{1\over 3}+ {1\over 5} -{1\over 7} +{1\over 9} - \cdots$$

• express functions $$\cos x = 1-{x^2\over 2} + {x^4\over 4!} -{x^6\over 6!} +{x^8\over 8!} - \cdots$$

• approximate a function using the first few terms

• represent a function as a polynomial with infinitely many terms (evaluate, differentiate, integrate), e.g., $\cos x$

Infinite sequences

An infinite sequence, or sequence is a list of numbers $$\{a_1,a_2,\cdots,a_n,\cdots\}$$

• A sequence is a function whose domain is the set of integers $\{1,2,3,\cdots\}$, i.e., $a_1=f(1)$, $a_2=f(2), \cdots.$

• The integer $n$ is called the index of $a_n$.

• The index set can start with any integer number other than 1, e.g., $\{a_0,a_1,a_2,\cdots\}$, $\{b_2,b_3,b_4,\cdots\}$, $\{c_{-2},c_{-1},c_0,c_1\cdots\}$ are sequences.

• We use $\{a_n\}_{n=2}^\infty$ to denote $\{a_2,a_3,a_4,\cdots\}$. Sometimes, we just say $\{a_n\}$ if the index set is not important or is clear from the context.

Example 1

Consider the sequence $-1,~4,~-9,~16,-25,\cdots$. It can be written in the following ways:

• $a_n=(-1)^nn^2$ for $n\geq 1$.

• $\{(-1)^nn^2\}$ (n starts at 1 by default)

• $\{(-1)^nn^2\}_{n=1}^\infty$

• $\{(-1)^n(n-2)^2\}_{n=3}^\infty$

• $\{-1,~4,~-9,~16,-25,\cdots\}$

A sequence can be defined recursively

Fibonacci sequence $$F_0=0,~F_1=1,~F_i=F_{i-1}+F_{i-2}, \quad \mbox{for } i\geq 2.$$

Graphing a sequence

Plot the first 100 terms of the following three sequence.

• Converges to $0$. $$1,{1\over2},{1\over3},{1\over4},\cdots,{1\over n},\cdots$$

• Diverges because it goes to infinity. $$1,2,3,4,\cdots,n,\cdots$$

• Diverges because it bounces back and forth. $$1,-1,1,-1,\cdots,(-1)^n,\cdots$$

Since there is no explicit form for the Fibonacci sequence, we define a function to compute the elements in the sequence.

Convergence

A sequence $\{a_n\}$ converges to the number $L$ if for any positive number $\epsilon$, there exists an integer $N$ such that for all $n>N$, $$|a_n - L|<\epsilon.$$

If $\{a_n\}$ converges to $L$, we write $\lim\limits_{n\rightarrow\infty} a_n = L$, or simply $a_n\rightarrow L$, and call $L$ the limit of the sequence $\{a_n \}$.

• $N$ depends on $\epsilon$, in general, a smaller $\epsilon$ requires a bigger $N$.

• $n>N$ can be replaced by $n\geq N$.

• If no such number $L$ exists, we say that $\{a_n\}$ diverges.

Examples

• For the case $$a_n={1\over n}.$$ We know that it converges to $0$, so $L=0$. We need to have $$|a_n-0|<\epsilon,$$ which is the same as $n>{1\over \epsilon}$. We we need to choose the integer $$N>{1\over \epsilon}.$$

• For the case $${n\over 2n+1}$$ We know that it converges to ${1\over 2}$, so $L={1\over 2}$. We need to have $$\left|a_n-{1\over 2}\right|={1\over 2(2n+1)}<\epsilon,$$ which is the same as $$n>{1\over2}\left({1\over 2\epsilon}-1\right).$$

• For the case $$(-1)^n$$ If $L$ exists, then we have $L\in (1-\epsilon, 1+\epsilon)$ and $L\in (-1-\epsilon, -1+\epsilon)$ for a small $\epsilon$, which can not happen. So $L$ does not exist.

Divergence to infinity

• $$\lim\limits_{n\rightarrow\infty}a_n=\infty, \mbox{ or } a_n\rightarrow \infty$$

If for every $M\in R$, there exists $N$ such that $a_n>M$ for all $n>N$. We say that $\{a_n\}$ diverges to $\infty$

• $$\lim\limits_{n\rightarrow\infty}a_n=-\infty, \mbox{ or } a_n\rightarrow -\infty$$

If for every $M\in R$, there exists $N$ such that $a_n>M$ for all $n>N$. We say that $\{a_n\}$ diverges to $-\infty$

Summary: Possible outcomes

• Convergence

• Divergence

  • Divergence to $\infty$ or $-\infty$.

  • Other divergence cases $$\{1,-2,3,-4,\cdots\}$$ $$\{1,0,2,0,3,0,\cdots\}$$

Remarks: The first finitely many terms do not change the convergence

Adding/removing/changing finitely many terms to a sequence does not change its convergence/divergence.

• $$\{7, 48, 56, -\pi, {1\over 2}, {1\over 3}, {1\over 4}, \cdots\}$$
• $$\{46, 3, e^{-1}, -1, 1, -1, 1, -1,\cdots\}$$

Calculating limits of sequences

Similar to the theorems on limits of functions in Chapter 2, we have the following theorem.

Let $\{a_n\}$ and $\{b_n\}$ be convergent sequences of real numbers with $a_n\rightarrow A$ and $b_n\rightarrow B$.

• Sum Rule: $\lim\limits_{n\rightarrow\infty}(a_n+b_n)=A+B$

• Difference Rule: $\lim\limits_{n\rightarrow\infty}(a_n-b_n)=A-B$

• Constant Multiple Rule: $\lim\limits_{n\rightarrow\infty}(k\cdot b_n)=k\cdot B$ (any real number $k$)

• Product Rule: $\lim\limits_{n\rightarrow\infty}(a_n\cdot b_n)=A\cdot B$

• Qutient Rule: $\lim\limits_{n\rightarrow\infty}{a_n\over b_n}={A\over B}$\quad if $B\neq 0$

Example

Note that $${n^2+1\over n^3+3}={\displaystyle{1\over n}+{1\over n^3}\over \displaystyle 1+{3\over n^3}}.$$ So $$\lim\limits_{n\rightarrow\infty}{n^2+1\over n^3+3}={0+0\over 1+0}=0.$$

Remark

• It can happen that $\lim\limits_{n\rightarrow\infty}(a_n+b_n)$ exists but both $\lim\limits_{n\rightarrow\infty}a_n$ and $\lim\limits_{n\rightarrow\infty}b_n$ do not exist.

  • Example: $$a_n = n + 1,\qquad b_n= 1-n$$

• $\{a_n\}$ and $\{ca_n\}$ ($c\neq 0$) are divergent or convergent at the same time.

Note: Limit treats the sequence as a function.

In the following example, the limit returns "ind", which means "indefinite but bounded." However, the sequence $\sin (\pi n)$ has a limit 0, but the function $\sin(\pi x)$ oscillates between 1 and -1 forever.

Instead, we can have the assumption that $n$ is integer by assume(n, 'integer')

Sandwich theorem for sequences

Let $\{a_n\}~,\{b_n\},$ and $\{c_n\}$ be sequences of real numbers. If $a_n\leq b_n\leq c_n$ holds for all $n\geq N$, and if $\lim\limits_{n\rightarrow\infty}a_n=\lim\limits_{n\rightarrow\infty}c_n=L$, then $\lim\limits_{n\rightarrow\infty}b_n=L$.

• $\lim\limits_{n\rightarrow\infty}{n!\over n^n}$

If $\lim\limits_{n\rightarrow\infty}|a_n|=0$, then $\lim\limits_{n\rightarrow\infty}a_n=0$

Continuous function theorem for sequences

Let $\{a_n\}$ be a sequence with $a_n\rightarrow L$ and $f$ is a function that is continuous at $L$ and defined at all $\{a_n\}$, then $f(a_n)\rightarrow f(L)$.

• Example: $\lim\limits_{n\rightarrow\infty} 2^{1\over n}$

Using l'Hopital's rule

Support that $f(x)$ is a function defined for all $x\geq n_0$ and that $\{a_n\}$ is a sequence such that $a_n=f(n)$ for all $n\geq n_0$. Then $$\lim\limits_{x\rightarrow\infty}f(x)=L\Longrightarrow\lim\limits_{n\rightarrow\infty}a_n=L.$$

• Example: $\lim\limits_{n\rightarrow\infty}{\ln n\over n}$

When $n\rightarrow \infty$, we have $$\lim\limits_{n\rightarrow\infty}{\ln n\over n}=\lim\limits_{n\rightarrow\infty}{{1\over n}\over 1}=0$$

$\lim\limits_{n\rightarrow\infty}{\left({n+1\over n-1}\right)^n}$

If we take the limit, we get $1^\infty$, which is undeterminated.

We can take the log first, we have

Therefore, $$\lim\limits_{n\rightarrow\infty}{\left({n+1\over n-1}\right)^n}=e^2$$

Find the limit of a recursively defined sequence

When we are given a recursively defined sequence, we can first look at the graph to see if the limit exists or not. Then we calculate many values of the sequence and see where we end up.

Example

If $\{a_n\}$ is the Fibonacci sequence, which does not converge, what about the sequence $$b_n={a_{n+1}\over a_n}.$$

We can take a look at the graph first.

The limit of the sequence is the "Golden Ratio" $${1+\sqrt{5}\over 2}.$$

Common limits ($x$ is fixed)

• $$\lim\limits_{n\rightarrow\infty}{\ln n\over n}=0$$
• $$\lim\limits_{n\rightarrow\infty}\sqrt[n]{n}=1$$
• $$\lim\limits_{n\rightarrow\infty}x^{1\over n}=1 \qquad (x>0)$$
• $$\lim\limits_{n\rightarrow\infty}x^n =0\qquad (|x|<1)$$
• $$\lim\limits_{n\rightarrow\infty}\left(1+{x\over n}\right)^n=e^x \qquad (\text{any }x)$$
• $$\lim\limits_{n\rightarrow\infty}{x^n\over n!}=0\qquad (\mbox{any }x)$$

Bounded sequences

A sequence $\{a_n\}$ is bounded from above if there exists a number $M$ such that $a_n\leq M$ for all $n$. The number M is an upper bound for $\{a_n\}$. If $M$ is an upper bound for $\{a_n\}$ but no number less than $M$ is an upper bound for $\{a_n\}$, then $M$ is the least upper bound for $\{a_n\}$.

A sequence $\{a_n\}$ is bounded from below if there exists a number $m$ such that $a_n\geq m$ for all $n$. The number $m$ is a lower bound for $\{a_n\}$. If $m$ is a lower bound for $\{a_n\}$ but no number greater than $m$ is a lower bound for $\{a_n\}$, then $m$ is the greatest lower bound for $\{a_n\}$.

If $\{a_n\}$ is bounded from above and below, then $\{a_n\}$ is bounded. If $\{a_n\}$ is not bounded, then we say that $\{a_n\}$ is an unbounded sequence.

• $\{n\}$

• $\{{n\over n+1}\}$

Bounded monotonic sequences

A sequence $\{a_n\}$ is nondecreasing if $a_n\leq a_{n+1}$ for all $n$. That is, $a_1\leq a_2\leq a_3\leq\cdots$. The sequence is nonincreasing if $a_n\geq a_{n+1}$ for all $n$. The sequence $\{a_n\}$ is monotonic if it is either nondecreasing or nonincreasing.

• $\{3\}$ is bounded monotonic

• ${(-1)^n}$ is bounded but not monotonic

• $\{{n\over n+1}\}$ is bounded and monotonic

A bounded and monotonic sequence converges.

Code for ploting a sequence and find the limit of a sequence