COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
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CHAPTER 4 SEQUENCES, JULIA and MANDELBROT SETS, and Power Series

Section 4.1 Sequences and Series

In formal terms, a complex sequence is a function whose domain is the positive integers and whose range is a subset of the complex numbers. For convenience, we at times use the term sequence rather than c omplex sequence .

If we wish a function to represent an arbitrary sequence, we could specify it by writing , , and so on. The values , , , ..., are called the terms of a sequence, and mathematicians, being generally lazy when it comes to things like this, often refer to , , , etc., as the sequence itself, even though they are really speaking of the range of the sequence when they do this. Mathematicians are also not so fussy about starting a sequence at , so that , , , ..., etc., would also be acceptable notation, provided all terms were defined.

Definition 4.1: Limit of a sequence

The expression

means that

for any real number >0, there corresponds a positive integer which depends on such that ( ) whenever > .

Theorem 4.1 Let and . Then,

if and only if and .

Example 4.1, Page 128. Find .

> L:='L': n:='n': Xn:='Xn': Yn:='Yn': Zn:='Zn': z:='z':
Xn := sqrt(n)/n:
Yn := (n+1)/n:
Zn := Xn + I*Yn:
L := limit(Zn, n=infinity):
z[n] = Zn; ` `;
limit( z[n], n=infinity) = L;

Example 4.2, Page 128. Show that the sequence diverges.

> n:='n': L:='L': n:='n': Zn:='Zn': z:='z':
Zn := (1 + I)^n:
L := limit(Zn, n=infinity):
z[n] = Zn; ` `;
limit( z[n], n=infinity) = L;

Maple did NOT find the limit!

Now look at the real and imaginary parts of .

> L:='L': n:='n': Xn:='Xn': Yn:='Yn': Zn:='Zn':
x:='x': y:='y': z:='z':
Xn := sqrt(2)^n * cos(n*pi/4):
Yn := sqrt(2)^n * sin(n*pi/4):
Zn := Xn + I*Yn:
L := limit(Xn, n=infinity) + I*limit(Yn, n=infinity):
`The general term is:`;
z[n], ` = `,x[n] + y[n] = Zn; ` `;
limit( z[n], n=infinity) = L;

Theorem 4.2 If is a Cauchy sequence, then converges.

Definition 4.3: Infinite Series

The formal expression is called an infinite series , and , , etc., are called the terms of the series.

If there is a complex number for which

= = ,

we will say that the infinite series converges to , and that is the sum of the infinite series.

When this happens, we write .

Theorem 4.3 Let and . Then

= =

if and only if and .

Theorem 4.4 (n-th term test)

If is a convergent complex series, then .

Example 4.3, Page 130. Show that the series converges.

> n:='n': S:='S': Xn:='Xn': Yn:='Yn': Zn:='Zn': z:='z':
Xn := 1/n^2:
Yn := (-1)^n/n:
Zn := Xn + I*Yn:
z[n] = Zn;
S := sum(Xn, n=1..infinity) + I*sum(Yn, n=1..infinity):
`Find the sum of the series:`;
Sum(z[n], n=1..infinity) =
Sum(Xn, n=1..infinity)+ I*Sum(Yn, n=1..infinity);
Sum(z[n], n=1..infinity) = S;

>

Example 4.4, Page 130. Show that the series diverges.

> n:='n': S:='S': Xn:='Xn': Yn:='Yn': Zn:='Zn': z:='z':
Xn := (-1)^n /n:
Yn := 1/n:
Zn := Xn + I*Yn:
z[n] = Zn;
S := sum(Xn, n=1..infinity) + I*sum(Yn, n=1..infinity):
`Find the sum of the series:`;
Sum(z[n], n=1..infinity) =
Sum(Xn, n=1..infinity)+ I*Sum(Yn, n=1..infinity);
Sum(z[n], n=1..infinity) = S;

Example 4.5, Page 131. Show that the series diverges.

> n:='n': S:='S': Zn:='Zn': z:='z':
Zn := (1 + i)^n:
z[n] = Zn;
S := sum(Zn, n=1..infinity):
`Find the sum of the series:`;
Sum(z[n], n=1..infinity) = Sum(Zn, n=1..infinity);
Sum(z[n], n=1..infinity) = S;
`But this is NOT the right answer !`;

BAD news! MAPLE substituted into a divergent geometric series ! It substituted into the
formula . You should always check to see if , or risk getting a wrong answer!

> n:='n': S:='S': Z:='Z': Zn:='Zn': z:='z':
Zn := (1 + I)^n:
Z := 1 + I:
S := 1/(1-Z) - 1:
`1/(1-Z) - 1 ` = S;
`where Z ` = Z;
`The general term is:`;
z[n] = Zn; ` `;
`But`;
1 < `|1+I| `;
1 < abs(Z);
1 < evalf(abs(Z));
`Therefore the series diverges.`;

End of Section 4.1.