Infinity and Beyond 5: Examples

Infinite sums

1. Geometric series: ∑n≥02−n
   20+2−1+2−2+2−3+…=2.
2. Telescoping series: ∑n≥11n(n+1)
   11⋅2+12⋅3+13⋅4+14⋅5+…=1.
3. James Gregory's (or Leibniz) series
   11−13+15−17+…=π4.
4. Euler's series: ∑n≥1n−2
   112+122+132+142+⋯=π26.
5. Euler's series: ∑n≥1(2n−1)−2
   112+132+152+172+⋯=π28.
6. Euler's alternating series: ∑n≥1(−1)n+1n−2
   1−122+132−142+152−162+⋯=π212.
7. Euler's alternating series: ∑n≥1(−1)n+1(2n−1)−3
   1−133+153−173+193−1113+⋯=π232.
8. Alternating Harmonic Series:
   11−12+13−14+…=ln(2).
9. Nilakantha (15th century) I:
   115+4⋅1−135+4⋅3+155+4⋅5−175+4⋅7+…=π16.
10. Nilakantha (15th century) II:
    3+433−3−453−5+473−7−493−9+…=π.

Infinite products

1. John Wallis' product
   2⋅2⋅4⋅4⋅6⋅6⋅…1⋅3⋅3⋅5⋅5⋅7⋅…=π2.
2. François Viète's product
   12⋅2√2⋅2+2√√2⋅2+2+2√√√2⋅2+2+22√√√√2⋅…=1π.
3. ∏n≥2(1−n−2)
   (1−122)⋅(1−132)⋅(1−142)⋅…=12.
4. ∏n≥3(1−4n−2)
   (1−432)⋅(1−442)⋅(1−452)⋅…=16.

Continued fractions

1. π
   1+122+322+522+722+…=4π.
2. more π
   1+123+225+327+429+…=4π.
3. Golden ratio
   1+11+11+11+…=ϕ=1+5√2.

What not

1. Golden ratio
   1+1+1+1√+…−−−−−−−−−−√−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−√=ϕ=1+5√2.

Infinity and Beyond 4: Sums

One of the first ways we experience infinity is through sums.

1+1+1+1+1 ... = ∞

Any sum that goes out to infinity is said to DIVERGE.  Let's look at some examples

HARMONIC SERIES:

1 + 1/2 + 1/3 + 1/4 + ... = ∞

Proof: 

1 + (1/2) + (1/3+1/4) + (1/5+1/6+1/7+1/8) + (1/9+1/10+1/11+1/12+1/13+1/14+1/15+1/16) + ...

Each sub-sum is greater than or equal to 1/2, so the sum is greater than

1 + 1/2 + 1/2 + 1/2 + ... = ∞

Power of two series:

1 + 1/2 + 1/4 + 1/8 + ... = 2

This can be seen by cutting a line segment over and over again (One piece will have size 1, the next 1/2, etc)


A very famous theorem says that if we define:

Z(s) = 1^-s + 2^-s + 3^-s + 4^-s + ...

Z(s) diverges when s < 2.

(The next post will give more examples of converging series)

Here is a fun problem using the above setup:

The tail of a giant wallaby is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the wallaby (hungrily). The wallaby sees the flea leaps into the air and lands one mile from the stake (with its tail still attached to the stake by the rubber band). The flea does not give up the chase but leaps into the air and lands on the stretched rubber band one inch from the stake. The giant wallaby, seeing this, again leaps into the air and lands another mile from the stake (i.e., a total of two miles from the stake). The flea is undaunted and leaps into the air again, landing on the rubber band one inch further along. Once again the giant wallaby jumps another mile. The flea again leaps bravely into the air and lands another inch along the rubber band. If this continues indefinitely, will the flea ever catch the wallaby? (Assume the earth is flat and continues indefinitely in all directions.)

Answer:

For this question, I defined a unit to be the ratio between a mile and an inch.  When the flea first jumps one inch onto the rubber band, for every mile the wallaby jumps, the flea is pulled forward one inch.  We can therefore say that the flea has moved forward one unit.  Now, when the flea jumps another inch, for every two miles the wallaby jumps, that jump will expand one inch.  We can say the flea jumped a ½ unit.  Using this concept, we can say, when the flea jumps 

1 + 1/2 + 1/3 + 1/4 + ... 

units, does the flea reach 63360 units?  Because the harmonic series diverges, the answer is YES

Infinity and Beyond 3: Limits

The most likely source for the question whether 1/0 = ∞ is a realization that dividing 1 by ever smaller numbers produces numbers arbitrary large. In this context, ∞ is understood as a very big, in fact, even bigger than any other, number. In a sense, this is a good idea that may be worked out rigorously. However, the approach is not without pitfalls. One can say that ∞ is more "big" than "a number". This is because, no definition may make ∞ possess properties of (or behave like) all other numbers.

For example, assuming that indeed 1/0 = ∞, we should also accept (by exactly same reasoning) that 2/0 = ∞ implying that 2×∞ = ∞. Obviously, this is a property that is not shared by any real number. Similarly, ∞ + 1 = ∞ which after2×∞ = ∞ should not come as a surprise for the latter may be expected to mean ∞ + ∞ = ∞. So if adding another infinity does not change it, adding a mere 1 should not change it either.

On the up side, if 1/0 = ∞, then it is quite likely that 1/∞ = 0. Indeed, if we understand that 1/∞ is a substitute for dividing 1 by ever larger numbers, then 1/∞ may sensibly stand for a non-negative number which is smaller than any positive number; and 0 quite fits the bill.

There is one problem, though. 1/0 is an ambiguous expression. Somehow, 1/∞ = 0 makes more sense than 1/0 = ∞. The reason is that, the way it was used so far, ∞ may be approached from only one direction, viz. by letting a number grow without bound. Zero, on the other hand, may be approached from two directions. If 1 is divided by ever decreasing numbers the result grows without bound, as expected. But zero is exactly midway between positive and negative numbers and may be as easily approached by negative numbers decreasing in magnitude. The result will be a negative number whose magnitude grows without bound. This one is more appropriately denoted as -∞.

So what is ∞?

First of all, it is just a symbol for the concept of growing without bound. Instead of saying "let x (or n) grow without bound", mathematicians often say "let x (or n) tend to infinity" or "as x (or n) tends to infinity". There is a special shorthand for this, too: x → ∞ (or n → ∞).

As x → ∞, other quantities that depend on x, like say, f(x), may exhibit all kinds of behaviors. Some, like f(x) = x², will grow without bound. In such cases, we write

f(x) → ∞ as x → ∞,

or, introducing another symbol: lim,

lim f(x) = ∞ as x → ∞.

And also

limx → ∞ f(x) = ∞.

A quantity, f(x), dependent on x may grow without bound as x tends to a real number as well. In this case, we write

limx → a f(x) = ∞,

where a is a plain real number. In particular, if f(x) = 1/x, we would like to write

limx → 0 1/x = ∞.

However, as we already discussed, this rather meaningless, for 0 may be approached from two directions producing quite distinct results. Instead we use

limx → 0+ 1/x = ∞ and
limx → 0- 1/x = -∞

to distinguish between the two cases.

It is important to realize that none of the above makes ∞ a (real) number. In the real number system, 1/0 is quite meaningless, or, at best, ambiguous. Limits are studied at the beginning Calculus courses where it is shown that iff(x) → A and g(x) → B, as x → a, then

limx → af(x)g(x) = AB = (limx → af(x)) (limx → ag(x)).

However, taking g(x) = x, h(x) = x² and k(x) = √x, all of which grow without bound as x → ∞, and f(x) = 1/x, we see that

limx → ∞ f(x)g(x) = 1,
limx → ∞ f(x)h(x) = ∞, and
limx → ∞ f(x)k(x) = 0.

This tells us that the expression 0·∞ will forever remain undefined.

The addition of limits is handled similarly:

limx → a(f(x) ± g(x)) = A ± B = limx → af(x) ± limx → ag(x).

As with the product, it is not always possible to use that formula with infinite limits. An expression, ∞ - ∞ (in the sense of the difference of limits) may happen to evaluate to -∞, a finite number, or ∞ depending on the two limits involved. For f(x) = x and g(x) = x - sin(x), both limits are infinite:

limx → ∞f(x) = limx → ∞g(x) = ∞.

However, the difference f(x) - g(x) = sin(x) has no limit as x → ∞. Thus the expression ∞ - ∞ will also remain undefined.

For those curious, the symbol ∞ for infinity was borrowed from the Latin numeral 1000 by John Wallis in 1655. The symbol ∞ closely resembles the shape of lemniscate - a simple endless curve.

Infinity and Beyond 2: Finite sets

Finite is the opposite of infinite; something is finite if it's not infinite. But there are many different notions for infinity; there are several for the idea of finite.

In geometry, a set may or may not be contained in a finite portion of the plane (or space). A portion of the plane is finite, if it's contained in a ball - however big. In other words, a portion of the plane is finite if the set of all distances from its points to a fixed point (say, origin) is bounded. A figure lying in a finite portion of the plane is said to be bounded. A segment is a bounded - also and frequently finite - portion of an infinite line. This is so, even if a segment contains an infinite number of points.

There are also Finite Geometries that contain a finite number of points and lines.

The set rational numbers between 0 and 1 belongs to a finite segment but, in itself, is infinite.

Among numbers, the notion of finiteness is an outgrowth of our ability to count. Roughly speaking, a set of objects is finite if it can be counted.

The numbers 1, 2, 3, ... are known as "counting" just because this is what we do while counting: we call the names of those numbers one at a time while pointing (even if mentally) to members of a set. The last number to be called is the cardinality of the set. A set of cardinality N is in a one to one correspondence with the set {1, 2, ..., N}.

Thus a set is finite if its cardinality is an integer. A set is infinite if it is not finite.

We may establish another definition of infinitude as a consequence of what has been said so far

Theorem

A set is infinite if and only if it contains a proper subset of the same cardinality.

(The empty set Ø is considered finite as well - it is certainly does not appear infinite.)

Proof

The proof will emerge in a succession of Lemmas.

Adding an element to a finite set leaves a finite set.

Let there be set A of cardinality N: |A| = N. The elements of A are in a 1-1 correspondence with the integers{1, 2, ..., N}. Adding an element to A increases its cardinality by 1, for the 1-1 correspondence between A and{1, 2, ..., N} is naturally expanded to augmented set and {1, 2, ..., N, N+1}. Just assign N+1 to the new element.

But N + 1 is a counting number as much as N itself. Therefore, the augmented set is also of finite cardinality, i.e., is finite.

Removing and element from an infinite set leaves an infinite set.

Indeed, if we remove an element from an infinite set, the remaining set is bound to be infinite; for, otherwise, putting that element back in we would get a finite set.

The set N = {1, 2, 3, ...} of natural numbers is infinite.

Assuming that |N| = k, for an integer k, leads to a contradiction because already |{1, 2, ..., k, k+1}| = k+1 so that counting elements of N we can reach k+1 which is greater than k.

A set whose cardinality equals that of N is said to be countable and is called a sequence.

Every infinite set contains a sequence.

Just keep removing one element at a time.

A sequence contains a subset of the same cardinality.

Indeed, this is true of N which is equivalent to the set of odd numbers, the set of even numbers, the set of squares - and what not. For what follows one example is especially handy. The 1-1 correspondence n→n+1 shows that N and {2, 3, ...} have the same cardinality.

Any infinite set contains a set of the same cardinality.

Let set A contains a sequence S. If A = S, we are finished. Otherwise, let T = A - S, and s ∈ S. There exist a 1-1 correspondence between S and S-{s}. This correspondence is expanded to a correspondence between A and A-{s} by assigning elements from T to themselves.

Finally,

Any subset of a finite set is finite.

and

A superset of an infinite set is infinite.

Infinity and Beyond 1: What is Infinity?

One of the most difficult questions a curious student may ask a math teacher is whether 1/0 = ∞ or not. And then of course comes another thoughtful extension: Is 1/∞ = 0?

Why this is a difficult question? It is difficult because it is usually asked by those who feel that infinity is a natural concept, like a number is, and everyone, especially a math teacher, should not have difficulty answering the question. But, for one, it is hard to insist that even number is a natural concept. Indeed, while counting numbers have been realized by various cultures, zero and negative numbers were very long in coming, not to mention decimals and complex numbers.

Secondly, even for those numbers that are perceived natural, the concept of division is not fundamental and has to be defined in the course of a study. Addition for natural numbers is an outgrowth of counting and may be easy to define (or accept) for the numbers 1, 2, 3, ... But all the rest (meaning other numbers and operations, like subtraction, multiplication, division) require a definition. So, in the absence of any preliminary understanding or common knowledge, the best (but seldom expected or acceptable) answer is, What do you mean by ∞ or, for that matter, by dividing by 0?

Clearly, at the beginning, intuition played a more important role than rigor. After Leibniz's death, it took about 250 years to set Calculus on a reasonably solid foundation. We are now about 150 year past this landmark, at the time when the standards of mathematical thought and education have evolved dramatically. To be understood by others, it is imperative to fall in line and use the common language.

Thirdly, even assuming there is a useful definition of infinity that deserves a recognition and a symbol of its own, it is not yet obvious that for this infinity it is possible in a reasonable manner to define arithmetic operations with more common numbers.

Finally, there are many infinities in mathematics; and this is true in more than one sense. There are many infinities and, say, ∞ is not a common notation used to denote each of them. Besides ∞, other symbols, for example,  and ω, are in circulation that denote infinities very much different from the one (or ones?) that ∞ usually stands for. Various infinities are defined differently and are subject to different operations and different laws. For example, while  + 1 = 1 + , ω + 1 ≠ 1 + ω.

It is prudent then to deal with various infinities one at a time. We shall look at several. For many of those, the arithmetic operations make no sense. For others, they do, but the definitions and the results differ.

Infinity and Beyond

I am starting a new series on Infinity and Beyond.  Leave ideas in the comments