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

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

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

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 1.586.18 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!

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...

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...

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,

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 ≈ 1.4447 the largest convergent value of x

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 0.20.2 is the 5th root of 0.2 and hence closer to 1 0.20.7248 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

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 0.9472 + 0.3208i Therefore:

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

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

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

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