# Topic 11 Infinite Sequences and Series

## 11.1 Introduction to Sequences and Series

A sequence is a list of numbers in a definite order (indexed by integers). A series may be considered as the limit of the sequence of partial sums.

When the sequence is explicitly defined by an mathematical expression \(a_n=f(n)\), Maple has the following command to list numbers of the sequence `seq(f, i=m..n, step)`

.

**Example 11.1 **
Find the first 10 terms of the sequence \(\{\frac{1}{n(n+1)}\}_{n=1}^\infty\). Determine whether the sequence \(\{\frac{1}{n(n+1)}\}\) is convergent or divergent.

*Solution. *

```
# using seq
seq(1/(n*(n+1)), n=1..10)
```

The sequence converges to \(0\).

For a series \(\sum a_n\), normally it is not easy to find explicit expression for the partial sum \(s_n=\sum\limits_{k=1}^n a_k\). However, if sequence is defined by an mathematical expression \(a_n=f(n)\), we may find values of partial sums recursively use a `for/from loop`

statement in Maple:

```
for *counter* from *initial* by *increment* to *final* do
statement_sequence;
end do;
```

**Example 11.2 **
Find the first 20 partial sums \(s_k=\sum_{n=1}^{n}a_n\) of the infinite series
\[
\sum_{n=0}^\infty\frac1{2^n}=1+\frac12+\frac14+\frac18+\cdots.
\]
Determine whether the series \(\sum_{n=0}^\infty\frac1{2^n}\) is convergent or divergent.

*Solution. *

```
# Set up s when n=0
s:=1
# Find 10 terms using `for/from loop`
for n from 1 to 10 do
s:=s+1/(2^n);
end do;
```

The series converges to 2.

Of course, we may also use `for/from loop`

to list numbers of a sequence.

*Solution. * Second solution to example 11.1.

```
# using `for/from loop`
for n from 1 to 10 do
1/(n*(n+1);
end do;
```

When the sequence is defined by a recurrence formula like the Fibonacci sequence, we will need to Maple how to interpret the formula. For that purpose, we use a procedure, which encloses a sequence of statements between `proc(...)`

and `end proc`

, to define the formula in Maple.

For example, the following is a procedure that defines a function \(a(x)=\sqrt{x}-\frac{1}{\sqrt{x}}\):

`a:=proc(x) sqrt(x)-1/sqrt{x}; end proc;`

To structure codes in a procedure, you may use `Code Edit Region`

which can be find in the `Insert menu`

.
To execute codes within this region, click `Execute Code`

from the `Edit`

menu, or use the shortcut command `Ctrl+E`

.

**Example 11.3 **
The Fibonacci sequence is defined by \(fib(0)=0\), \(fib(1)=1\) and \(fib(n)=fib(n-1)+fic(n-2)\).
Find the first 20 Fibonacci numbers.

*Solution. *
We first define a function \(fib(n)\) which returns the \(n\)-th Fibonacci number.

```
fib := proc (n::nonnegint)
if 2 <= n then
return fib(n-1)+fib(n-2):
else
return n:
end if;
end proc
```

Now we can use either `seq()`

or `for/from loop`

.

seq(fib(n), n=0..19)

**Exercise 11.1 **
Find the first 20 terms of the sequence
\[
\{\sin{\frac{\pi}{n}}\}_{n=1}^\infty.
\]
Determine whether the sequence \(\{\sin{\frac{\pi}{n}}\}\) is convergent or divergent.

**Exercise 11.2 **
Find the first 20 partial sums \(s_k=\sum_{n=1}^{n}a_n\) of the infinite series
\[\sum_{n=0}^\infty\frac1n=1+\frac12+\frac13+\frac14+\cdots.
\]
Determine whether the series \(\sum_{n=0}^\infty\frac1n\) is convergent or divergent.

**Exercise 11.3 **
Find the 20th to 30th Fibonacci numbers.

## 11.2 Power Series

A power series is a series with a variable \(x\): \[ \sum\limits_{n=0}^{\infty} c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+\cdots. \]

More generally, a series of the form

\[\begin{equation} \sum\limits_{n=0}^{\infty} c_n(x-a)^n=c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+\cdots \tag{11.1} \end{equation}\]

is called a power series at \(a\).

We call a positive number \(R\) the radius of convergence of the power series (11.1) if the power series converges whenever \(\left|x-a\right|<R\) and diverges whenever \(\left|x-a\right|>R\).

If a function \(f\) has a power series representation, i.e. \[ f(x)=\sum_{n=0}^\infty c_n(x-a)^n,\quad\quad \left|x-a\right|<R, \] then its coefficients are given by \(c_n=\dfrac{f^{(n)}(a)}{n!}.\)

**Example 11.4 **
Find the interval of convergence of the power series
\[
\sum\limits_{n=1}^{\infty}\dfrac{(-2)^nx^n}{n^3}.
\]

*Solution. *

```
# Find the abs(a_{n+1}/a_n)
q:=abs((-2)^(n+1)(n+1)^3/(-2)^(n+1)(n+1)^3);
# Find the limit of q
r:=limit(simplify(q), n=infinity)
# Find the interval of convergence
solve(abs(x)<1/r, x)
```

**Example 11.5 **
Find the Taylor expansion of the function \(f(x)=\dfrac{1}{x-2}\) at \(x=0\) up to the 5-th order. Plot \(f(x)\) and the \(5\)-th order Taylor polynomial together.

*Solution. *

```
# Find the Taylor expansion.
ftaylor:=taylor(1/(x-2), x = 0, 5)
# convert the Taylor series into a polynomial
fpoly:=convert(ftaylor, polynom)
# Plot the functions
plot([1/(x-2), fpoly], x=-1..1)
```

**Exercise 11.4 **
Find the interval of convergence of the power series
\[
\sum\limits_{n=1}^{\infty}\dfrac{(-4)^nx^n}{\sqrt{n}}.
\]

**Exercise 11.5 **
Find the Taylor expansion of the function \(f(x)=\sin x\) at \(x=0\) up to the 5-th order. Plot \(f(x)\) and the \(5\)-th order Taylor polynomial together over the interval \([-\pi,\pi]\).

## 11.3 Taylor Expansion

Let \(f(x)\) be a function. Assume that the \(k\)-th order derivatives \(f^k(a)\) exist for \(k=1, 2, \dots, n\). The polynomial \[ T_n(x)=\sum_{k=0}^n\dfrac{f^{(n)}(a)}{k!}(x-a)^k \] is called the \(n\)-th degree Taylor polynomial of \(f\) at \(a\).

Let \(f(x)\) be a function has derivative at \(a\) up to all orders. Set \[ R_n(x)=\sum_{k=n+1}^\infty \dfrac{f^{(k)}(a)}{k!}(x-a)^k,\quad\quad \left|x-a\right|<R, \] which is called the reminder of the Taylor series \[ \sum\limits_{k=0}^\infty \dfrac{f^{(k)}(a)}{k!}(x-a)^k. \]

If \[ \lim\limits_{n\to\infty} R_n(x)=0 \] for \(\left|x-a\right|<R\), then \(f(x)\) is the sum of the Taylor series on the interval, that is \[ f(x)=\sum\limits_{k=0}^\infty \dfrac{f^{(k)}(a)}{k!}(x-a)^k,\quad\quad \left|x-a\right|<R. \]

If \(\left|f^{n+1}{x}\right|\leq M\) for \(\left|x-a\right|\leq d\), then the reminder \(R_n\) satisfies the follow inequality \[ \left|R_n(x)\right|\leq \dfrac{M}{n+1}\left|x-a\right|^{n+1}\quad \text{for}\quad \left|x-a\right|\leq d. \]

Roughly speaking, the absolute value of the reminder \(\left|R_n(x)\right|\) determines how accurate the Taylor polynomial approximation.

**Example 11.6 **
Approximate function \(f(x)=\sin x\) by the degree 3 Taylor polynomial at \(x=1\).

*Solution. *

```
# Find the Taylor series.
fTs:=taylor(sin(x), x = 0, 4)
# Convert the Taylor series into a polynomial
fTp:=convert(fTs, polynom)
# Evaluate the Taylor polynomial at 1
subs(x=1, fpolyapprox)
```

**Example 11.7 **
Plot the function
\[
g(x)=\begin{cases}e^{-\frac{1}{x^2}} & x\neq 0\\ 0 & x=0\end{cases}
\]
and its 5-th order Taylor polynomial over the domain \([-2..2]\). What can you conclude?

*Solution. *

```
# Define a piece-wisely defined function.
g:=piecewise(x!=0, exp(-1/x^2), 0)
# Find Taylor polynomial of degree 5.
for n to 5 do T := (eval(diff(g(x), x$n), x = 0))*x^n/factorial(n)+T end do
# Plot the functions
plot([g,T],x=-2..2, color=[red, blue])
```

The graphs of the functions are shown in the picture.

In the solution, `x$n`

is a shortcut option for `x, x, x, x, x`

in the `diff`

command.

**Exercise 11.6 **
Approximate function \(f(x)=e^x\) by the degree 5 Taylor polynomial at \(x=1\).

**Exercise 11.7 **
Compare the function \(y=\sin x\) with its degree 10 Taylor polynomial at \(x=0\).