A Discussion on Infinity

Understanding Infinity

Many people think of infinity as a number, but it's actually a concept of something without end—a perpetual, endless quality. For example, an infinitely long road is one that continues endlessly. While we can use the idea of infinity in our thoughts, it's not something you can encounter in a literal or practical sense. For instance, if you were stranded on an island and saw water all around you, you might say the sea continues forever, even though you know it does eventually end somewhere.

In physics, we sometimes use infinity to simplify very large numbers. We say the sun's rays are parallel to each other because the distance between the sun and Earth is so vast that it's considered effectively infinite. We do this because we know that light rays coming from an infinite distance are parallel, which makes calculations much easier.

Let's consider a hypothetical scenario: an infinitely long train travels along an infinitely long track. How long would it actually take for the train to cover the entire distance of the track? This question highlights a fundamental point about infinity. When you encounter mathematical expressions like infinity multiplied by infinity or infinity divided by infinity, there's no single, defined answer. These are called indeterminate forms, and because they are not logically or mathematically defined, a specific and unique answer to this kind of question simply doesn't exist.

Some confusions whilst dealing with infinite series

Let's consider the following series,

\begin{aligned} S_1 &= 1+1+1+1+1+1+1+\cdots \\ S_2 &= 1-1+1-1+1-1+1-\cdots \\ S_3 &= 1+2+3+4+5+6+7+\cdots \\ S_4 &= 2+4+6+8+10+12+14+\cdots \\ S_5 &= 1+3+5+7+9+11+13+\cdots \\ S_6 &= 1-2+3-4+5-6+7-\cdots \end{aligned}

Let's explore this with a seemingly absurd application of arithmetic,

\begin{aligned} S_3 - S_3 &= (1 + 2 + 3 + 4 + \cdots) - (1 + 2 + 3 + 4 + \cdots) = 0 \end{aligned}

If we apply this term by term, we get:

\begin{aligned} S_3 - S_3 &= (1-1) + (2-2) + (3-3) + (4-4) + \cdots \\ 0 &= 0 + 0 + 0 + 0 + \cdots \end{aligned}

This seems to be consistent, but let's try a different approach which is shown in the text. Let's see what happens if we subtract the series in a different way:

\begin{aligned} S_3 &= 1 + 2 + 3 + 4 + 5 + 6 + 7 + \cdots \\ -S_3 &= \;\; - (1 + 2 + 3 + 4 + 5 + 6 + 7 + \cdots) \\ \implies 0 &= 1+1+1+1+1+1+1+\cdots \\ \implies & \boxed{S_1=0} \end{aligned}

Derivation of the Sum of an Infinite Geometric Series

The method demonstrated above is a classic technique used to evaluate infinite series. A well-known and correct application of this method is to derive a formula for the sum of an infinite geometric series.

Consider the infinite geometric series:

\begin{aligned} S &= a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + \cdots \\ -rS &= \:\:\: - (ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + ar^7 + \cdots) \\ \implies S(1 - r) &= a \\ \implies \boxed{S = \frac{a}{1 - r}} \end{aligned}

Here’s what we did: We multiplied the entire series by \(r\), which effectively shifts every term one place to the right. When we subtracted the new series from the original, all terms cancelled out except the first term \(a\). This gave us a simple equation that we solved for \(S\), yielding the formula for the sum of an infinite geometric series.

Note: This formula is valid only when \( |r| < 1 \), ensuring the series converges.

Sly, one can see,

S_1+S_2 = 2+2+2+2+2+2+\cdots

which is the same thing as saying

\begin{aligned} S_1+S_2 &= 2(1+1+1+1+1+1+\cdots) \\ \implies S_1+S_2 &= 2S_1 \\ \implies S_2 &= S_1 \end{aligned}

Using the previous result, we get:

\boxed{S_2 = 0}

Now, let us look at \(S_3\),

\begin{aligned} S_3 &= 1+2+3+4+5+6+7+\cdots \\ &= 1+(1+1)+(1+1+1)+(1+1+1+1)+\cdots \\ &= 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+\cdots \\ &= S_1 \end{aligned}

which, using the previous result, gives us:

\boxed{S_3=0}

We can say this about \(S_4\),

\begin{aligned} S_4 &= 2(1+2+3+4+5+6+7+\cdots) \\ \implies S_4 &= 2S_3 \end{aligned}

Again using the previous result, we can write that:

\boxed{S_4=0}

And for \(S_5\):

\begin{aligned} S_4-S_5 &= 1+1+1+1+1+1+1+\cdots \\ \implies S_4-S_5 &= S_1 \end{aligned}

Using the previous results, we get:

\boxed{S_5=0}

If we look properly we can see the following pattern:

\begin{aligned} S_6-S_2 &= -1+2-3+4-5+6-7+\cdots \\ \implies S_6-S_2 &= -S_6 \end{aligned}

Using the previous result, we get:

\boxed{S_6=0}

Now, let's talk. Do you think this is correct?

Any sane and logical person would say this is definitely wrong. But where did we exactly go on the wrong track?

This happens because we are dealing with infinite series here and infinite series don't really behave like finite ones. Before dealing with any kind of infinite series we must first check if it is converging or diverging. Before checking we cannot do normal arithmetic to the series as they are not some number but rather a concept that is not precisely describable.

This is what mathematics is all about. It's about exploration and not about rote memorization. Sometimes it even gives ridiculous results, and we feel like, "Ah, finally found a place where mathematics doesn't work," but actually, math is never wrong. Either, in spite of being ridiculous, the result is correct, or you must have somewhere done something that defies the axioms of mathematics.

Let me give you one crazy result to blow your mind.

1+2+3+4+5+6+7+\cdots = - \frac{1}{12}

And believe me, this is correct. Even though every rule of checking if the series is divergent or convergent, it gives that this series is divergent. But still, it has been proven correct in every way. And every attempt to prove this wrong builds up a contradiction.