1. Convergent Sequences of Real and Complex Exponentials
Infinity Wars
Convergent Sequences of Real and Complex Exponentials
Introduce ourselves and mention MGA
David Vogel Jonathan Joe
Middle Georgia State University

2. Infinite Sequences of Exponentials
Infinite Series
Infinite Product
Infinite Exponential
Formal Definition
Each term is defined by setting the base, x, to the power of the previous term
An infinite series converges only if the terms approach zero as x goes infinity
For an infinite product to converge, the terms must approach 1
Under what conditions, might an infinite exponential converge?
With a infinite series or product, the individual terms vary
The value x in our infinitely-repeated exponential is unchanging
Properly looked at as a sequence
First term is x, second is xx, and the third is formed by raising x to the power xx
Can such a sequence converge?
Obviously or we wouldn’t be giving this talk

3. Convergent and Divergent Values
Clearly the sequence converges for x = 1
Every term equals 1
Sequence obviously diverges for x = 2
What about values between 1 and 2?
X1 = 2
X2 = 4
One clearly convergent value is x = 1
Almost as clear is that the sequence diverges for x = 2
Jonathan proved divergence by showing the terms are larger than the natural numbers
Values larger than 2 must also diverge
The interesting part of the number line therefore becomes the segment between 1 and 2
X3 = 16
X4 = 65,536
X5 ≈ 1019,728

4. Convergent or Non-convergent?
Look at sequence values for
x = 1.5 5 10 15 20 1 2 3 4 6 7 8 9 11 12
Term
Value
Actually, most of this region diverges as well
We looked at the first dozen terms of the sequence when x = 1.5
Things go smoothly at first, but after a12, the sequence explodes upward
Since 1.52 > 2, you can foresee that is huge, and the next term would be astronomical
We also find divergence for x = 1.45 (took 42 terms)
Are there any convergent values besides 1? Yes!

5. On the Other Hand... Sequence does converge for What is the limit of ?
Computing the first few terms, we see that they seem to approach the limit S = 2
Since , the limit of the sequence is indeed S = 2
Yes, if we go a little lower, to , the sequence is convergent
Does anyone know what this limit is?
Right, it converges to S = 2!
We used this as a question on our Math Olympics test several years ago, but the students got to use calculators, and 2 was one of five choices
Recall that ; values smaller than this also converge since every term is smaller
Some questions arise...

6. Questions
Are there values larger than for which the sequence converges?
Answer: YES
Does the sequence converge to arbitrarily large limits?
Answer: NO
Can we determine an upper bound for the values of that do converge?
We can answer these three questions (for real numbers at least)
As we’ll see next...

7. Finding the Limit By definition: Take the natural logarithm:
If the sequence converges,
Start with the definition
Taking the natural log of both sides
Assuming convergence, we get...
Now we substitute the limit value S for both an and an+1
This makes it easy to solve for x
Given a value for S, we can compute x
I can’t come up with a way to solve this equation symbolically for S as a function of x
Nevertheless, a graph of ln(S)/S vs. S gives us additional insight
Therefore,

8. Solution Curve We plot S vs. ln(S)/S To find the maximum
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 1 2 3 4 5 6 7 8 9 10 S
ln( x )
We plot S vs. ln(S)/S
ln(S)/S = ln(x)
To find the maximum
Take derivative
Set it equal to zero
Solve
Since ln(S)/S = ln(x) in this plot, vertical axis shows both ln(S)/S and ln(x)
If we pick a value for S (horizontal axis), we read ln(x) from the vertical axis
It’s easy then to compute x itself
The curve rises steeply, and then slowly decreases
The most interesting feature is that the curve has a maximum
We can compute the location of the maximum simply by differentiating (some of you will have already guessed)
The maximum occurs at the coordinates S = e, ln(x) = 1/e
Another point of interest is (2, ln( ))
What is the significance of this?
If we pick a value larger than e, a horizontal line drawn from the curve to the vertical axis, intersects the curve in two places
This would mean that the same base value x converges to two different limits, a logical inconsistency
S = e is the largest possible limit, and e1/e ≈ the largest convergent value of x

9. Curve also Predicts Convergence for 0 < x < 1
These first 30 terms clearly indicate convergence
Example: x = 0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 5 10 15 20 25 30
Term
Value
x = 0.2
The curve also indicates that values of x smaller than 1 (but larger than zero) should converge
In this case, the terms don’t grow monotonically, however
is the 5th root of 0.2 and hence closer to 1
is farther from 1, but the next term is then closer, and so on
Proving that one of these sequences converges would be slightly harder
Would probably have to group into sequences of alternate terms and then show that they converge to the same limit
When we try to compute exponential sequences involving a negative x, the sequence departs from real numbers
We now look at sequences involving complex values

10. Complex Exponentials Example:
What is the meaning of when both numbers are complex?
Example:
The complex logarithm is multi-valued, so we choose the principal branch
Everyone is familiar with Euler’s formula for complex powers of e, and you also know how to take powers of i
In trigonometry we also teach students how to compute iq/p, that is, rational powers of i
One can also set a complex number z to a complex power c
As with real numbers, accomplished by starting with the exponential function
Since the complex logarithm is multi-valued, one has to choose a specific branch—the principal branch is easiest
Remembering that c is complex, we use Euler’s formula and end up with
One can also compute the sine and cosine of a complex number; the exponential and the trig functions are all complex values
Example: Exponentiating i twice, e end up with i
Therefore:

11. Complex Exponentials Does converge? Yes, it does! Limit: z = i
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
Limit:
First we plot these three terms of the sequence
The fourth term moves back over toward the imaginary axis
Now we fill in subsequent terms and see that the values spiral in toward an apparent accumulation point—the sequence does seem to converge
Connecting the points with lines highlights the spiral pattern

12. Cyclical Non-Convergence
Some complex sequences don’t converge but are still bounded
Example:
Terms alternate among three individual accumulation points
1.5i
1.0i
0.5i
0.0i
When we pick a different value, however, the sequence does not converge but does not diverge to infinity either
Sequence terms alternate cyclically among three different centers
Graph includes an equal number of values near (0, 0) and (1, 0), but the spiral is tighter for these two accumulation points

13. Non-convergent Complex Values
Points in blue are divergent values
For what other values of z might the sequence
converge?
This graph and the points shown on the previous two slides were computer using a routine written in Python
This plot includes a grid of about 30,000 different values—the run time needed to compute all these sequences was about an hour
The points in blue indicate values for which the sequence does diverge (to infinity)
White figure-8 shaped area in the middle consists of points that converge to a single value
Remaining white areas—even those toward the upper and lower corners on the left—are points for which the sequence is bounded but alternates among different centers
Some display cycles of 3, 6, or 9
We believe other periods are also possible

14. Open Questions
Are there some rules that determine when a complex sequence is unbounded?
Can we predict why some complex sequences are cyclic, while others converge on a single limit point?
Can we prove mathematically that a given complex sequence is either convergent, bounded, or divergent?
Some questions for further investigation
We also need to examine real-number convergence more carefully
Proving convergence is relatively easy for the numbers between 1 and e1/e
Proving divergence for number between e1/e and 2 might be more difficult
Need to consider values smaller than 1—some may not converge to a single value