Lecture 2

Our goal here is to begin to learn to use Mathematica while at the same time use it to explore the

concepts in the text. You are encouraged to confirm the Mathematica results and the mechanical issue

of running the program, entering the commands, etc., i.e., run your own session of Mathematica and

enter these commands. Mathematica has pretty good built-in Help and you are strngly encouraged to

use it to look-up definitions of functions, find new functions, etc. The key stroke instructing Mathemat-

ica to perform a command (or evaluate an expression) is the combination shift-enter. The command to

open a text line to annotate your work is Alt-7.

From Eq. 2.5 we use the "sum" function in Mathematica to consider the (finite) sum

In[1]:= Sum@x^n, 8n, 4<D

Out[1]= x + x2

+ x3

+ x4

Alternatively we can enter the same expression with

In[2]:= Sum@x^n, 8n, 1, 4<D

Out[2]= x + x2

+ x3

+ x4

Note the general feature that Mathematica always begins the names of pre-installed functions with

capital letters, puts the arguments in square brackets, []'s, and uses curly brackets, {}'s, to define param-

eters for the function; here to sum over the index n from 1 to 4 (i.e., {n,4} = {n,1,4}).

Now define the infinite version (note how Mathematica likes to label the "quantity" infinity; it can also be

entered with the symbol ¥). In defining a function we use the underscore, _, to label the arguments (on

the left-hand-side) and the symbol ":=" (and not just "=")

In[3]:= S@x_D := Sum@x^n, 8n, Infinity<D

To ask Mathematica to perform the sum we simply type the name of the function (with no underscoring),

In[4]:= S@xD

Out[4]= -

x

-1 + x

This is a common result that Mathematica does not simplify automatically to the expected analytic

expression. If you expect that simplification is possible, you can ask Mathematica to look for it. There

is a handy shorthand if you want to perform an operation of the previous expression - just represent it

by the % symbol.

In[5]:= Simplify@%D

Out[5]= -

x

-1 + x

Given the current order in which the commands have been evaluated, this last command is the same as

In[6]:= Simplify@S@xDD

Out[6]= -

x

-1 + x

This the result of Eq.(2.10). For comparison we can also evaluate the sum starting with 1, the power

zero, as in Eq.(2.9), the so-called geometric series

This the result of Eq.(2.10). For comparison we can also evaluate the sum starting with 1, the power

zero, as in Eq.(2.9), the so-called geometric series

In[7]:= Sum@x^n, 8n, 0, 4<D

Out[7]= 1 + x + x2

+ x3

+ x4

In[8]:= Sum@x^n, 8n, 0, ¥<D

Out[8]=

1

1 - x

An essential strength of the Mathematica software is the ability to analytically perform this infinite

summation for a general variable x, as here. On the other hand you have to be careful not to be misled

by the result. In particular, for |x| < 1 the infinite series defines the function indicated. However, while

the functions is well defined for |x| > 1, the infinite series itself is divergent. We can study the issue of

convergence by looking at the behavior of the finite (truncated) series and then the remainder. First

define the finite series as a function of both x and N.

In[9]:= SN@N_, x_D := Sum@x^n, 8n, 1, N<D

In[10]:= SN@4, xD

Out[10]= x + x2

+ x3

+ x4

So let's use our plotting ability to check the behavior of this function as a function of N for fixed values

of x, e.g., x = 0.5.

In[11]:= Plot@SN@N, 0.5D, 8N, 1, 20<D

Out[11]=

10 15 20

0.85

0.90

0.95

1.00

Let's clean this up a bit and provide labels.

2 Lec2_227_08.nb

In[12]:= Plot@SN@N, 0.5D, 8N, 1, 20<, PlotRange ® 80, 1<,

AxesLabel ® 8N, SN<, AxesOrigin -> 80, 0<D

Out[12]=

0 5 10 15 20

N

0.2

0.4

0.6

0.8

1.0

SN

It clearly converges by N around 15. Now lets try a few values of x.

In[13]:= Plot@8SN@N, 0.3D, SN@N, 0.5D, SN@N, 0.7D, SN@N, 0.9D<, 8N, 1, 20<,

PlotRange ® 80, 1<, AxesLabel ® 8N, SN<, AxesOrigin -> 80, 0<D

Out[13]=

0 5 10 15 20

N

0.2

0.4

0.6

0.8

1.0

SN

Clearly both the number the series converges to varies with x and the actual converge properties vary

with the x value, but the details are difficult to see in this plot. To clean up the plot we extend the limits

and the axes.

Lec2_227_08.nb 3

In[14]:= Plot@8SN@N, 0.3D, SN@N, 0.5D, SN@N, 0.7D, SN@N, 0.9D<, 8N, 1, 50<,

PlotRange ® 80, 10<, AxesLabel ® 8N, SN<, AxesOrigin -> 80, 0<D

Out[14]=

0 10 20 30 40 50

N

2

4

6

8

10

SN

However, note that

In[15]:= Plot@SN@N, 1.1D, 8N, 1, 20<, PlotRange ® 80, 50<,

AxesLabel ® 8N, SN<, AxesOrigin -> 80, 0<D

Out[15]=

0 5 10 15 20

N

10

20

30

40

50

SN

So for |x| > 1 we see the finite sum diverging as N gets large. To focus on the question of convergence/-

divergence can look at the remainder of all the terms in the infinite series after the Nth term, RN = S -

SN.

In[16]:= RN@N_, x_D := Sum@x^n, 8n, 1 + N, Infinity<D

In[17]:= RN@N, xD

Out[17]=

x1+N

1 - x

In[18]:= SN@N, xD + RN@N, xD

Out[18]=

x1+N

1 - x

+

x H-1 + xNL-1 + x

A bit messy so simplify.

4 Lec2_227_08.nb

In[19]:= Simplify@SN@N, xD + RN@N, xDD

Out[19]= -

x

-1 + x

The expected analytic result for the full sum. Consider the remainder numerically.

In[20]:= RN@50, .3D

Out[20]= -2.22045 ´ 10-16

Mathematica has trouble recognizing the convergence but we can look for it graphically

In[21]:= Plot@RN@N, 0.3D, 8N, 1, 20<, PlotRange ® 80, 1<,

AxesLabel ® 8N, RN<, AxesOrigin -> 80, 0<D

Unset::wrsym : Symbol N is Protected. �

Out[21]=

0 5 10 15 20

N

0.2

0.4

0.6

0.8

1.0

RN

Note again that Mathematica struggles to perform the sum numerically, but still suggests that the

remainder shrinks with increasing N. Now consider the 4 x values used above

In[22]:= Plot@8RN@N, 0.3D, RN@N, 0.5D, RN@N, 0.7D, RN@N, 0.9D<, 8N, 1, 50<,

PlotRange ® 80, 1<, AxesLabel ® 8N, RN<, AxesOrigin -> 80, 0<D

Out[22]=

0 10 20 30 40 50

N

0.2

0.4

0.6

0.8

1.0

RN

As expected all the remainders head to zero for N®¥, although even on the current scale the remain-

der has only shrunk to about 0.05 at N = 50. Clearly this is not very satisfactory and is why, even with

computers available, we also need the analytic tests for convergence discussed in Lecture 2.

Lec2_227_08.nb 5

As expected all the remainders head to zero for N®¥, although even on the current scale the remain-

der has only shrunk to about 0.05 at N = 50. Clearly this is not very satisfactory and is why, even with

computers available, we also need the analytic tests for convergence discussed in Lecture 2.

What about the case x = 1

In[23]:= Plot@RN@N, 1.0D, 8N, 1, 20<, PlotRange ® 80, 1<,

AxesLabel ® 8N, RN<, AxesOrigin -> 80, 0<D

Sum::div : Sum does not converge. �

Sum::div : Sum does not converge. �

Sum::div : Sum does not converge. �

General::stop : Further output of Sum::div will be suppressed during this calculation. �

Out[23]=

0 5 10 15 20

N

0.2

0.4

0.6

0.8

1.0

RN

So in this case Mathematica knows that something is wrong.

Another way to look at is in terms of the limit, N ® ¥ (where we expect a zero answer for convergence)

In[24]:= Limit@RN@N, 0.5D, N ® ¥DOut[24]= 0.

So Mathematica does not handle this well. On the other hand once we have "summed" the general sum

In[25]:= RN@N, xD

Out[25]=

x1+N

1 - x

We can use this form to take the largeN limit for fixed x easily (note the substitution command /.x-

>value)

In[26]:=

x1+N

-1 + x

�. x ® 0.9

Out[26]= -10. 0.91+N

Take the limit

In[27]:= LimitB-

x1+N

-1 + x

�. x ® 0.9, N ® ¥F

Out[27]= 0.

So the series converges. At the boundary x = 1 we have

6 Lec2_227_08.nb

In[28]:= LimitB-

x1+N

-1 + x

�. x ® 1.0, N ® ¥F

Power::infy : Infinite expression

1

0.

encountered. �

Out[28]= ComplexInfinity

So this expression clearly diverges, and for x > 1

In[29]:= LimitB-

x1+N

-1 + x

�. x ® 1.1, N ® ¥F

Out[29]= -¥

We can also see this behavior in the plot of the summed series S[x], i.e., the divergence as x

approaches 1.

In[30]:= Plot@S@xD, 8x, -2, 2<, AxesLabel ® 8x, S<D

Sum::div : Sum does not converge. �

Sum::div : Sum does not converge. �

Sum::div : Sum does not converge. �

General::stop : Further output of Sum::div will be suppressed during this calculation. �

Out[30]=

-2 -1 1 2

x

1

2

3

4

5

S

So if we carefully evaluate the infinite series representing the remainder and takes its limit, N®¥ ,

Mathematica will accurately tell us about the convergence properties of the series. On the other hand,

we often have difficulty proceeding just numerically. The analytic analysis we have discussed in the

Lecture is essentially for proceeding either by hand or via Mathematica. To make use of series, we

need to understand when they make sense and when they do not via analytic methods.

Lec2_227_08.nb 7