Someone and Anyone

Sometimes, an infinite series may not be as expressive as (or carry as much information as) the function it represents. In this post, we’ll mainly discuss this concept (and also look at conditional and absolute convergence briefly).

Consider the following infinite series: f(x) = 1 + x + x² + x³ + x⁴ + x⁵ + …. Does this series ever converge? Try substituting 1/2 for x and see what happens. It converges, as we saw here. So, we write this as f(1/2) ~ 2. The twiddle or tilde sign here indicates that f(x) asymptotically tends to 2 at x = 1/2. Similarly, it is possible to prove the following: f(-1/2) ~ 2/3; f(1/3) ~ 3/2; f(-1/3) ~ 3/4 and so on. It is trivially true that f(0) = 1.

Let us now substitute 1 for x. We get f(1) = 1 + 1 + 1 + 1 + …. The series diverges at x = 1. It is obvious that the series diverges for x = 2, 3, 4, 5, … too. Observe the behaviour of f(x) when x = -1. We get f(-1) = 1 – 1 + 1 – 1 + 1 – 1 + …. If you take an even number of terms, then f(-1) = 0, and if you take an odd number of terms, then f(-1) = 1. We see that the partial sum of this series is definitely not becoming infinitely large. However, it is not converging either. This is considered a form of divergence. We can verify that f(-2), f(-3), f(-4), … also diverge. Loosely speaking, they seem to oscillate and go off to positive infinity and negative infinity at once.

We see that f(x) seems to have values only when x is between -1 and 1, exclusive. In other words, we have:

Observation 1: The domain of the function f(x) is from -1 to 1, exclusive.

Now, let us rewrite f(x) here and simplify it a bit.

f(x) = 1 + x + x² + x³ + x⁴ + x⁵ + …

That is, f(x) = 1 + x (1 + x + x² + x³ + x⁴ + … )

This implies f(x) = 1 + x f(x)

Therefore, we have f(x) = 1 / (1-x)

On simplification, we have:

Equation 1: 1 / (1-x) = 1 + x + x² + x³ + x⁴ + x⁵ + …

Now, the question we ask is: Are the LHS and the RHS of this equation one and the same? Earlier in this post, we had discussed the value of the RHS for various values of x, viz. -1/2, -1/3, 0, 1/3, 1/2. You will see that the LHS concurs. However, they are not one and the same thing! They have different domains.

We can see that 1 / (1-x) has values everywhere except at x = 1. When we see this in contrast to Observation 1, we see that the domain of 1 / (1-x) is “stretched”. It includes the domain of the infinite series and more. This indicates that an infinite series sometimes defines only a part of a function. More appropriately, an infinite series might define a function over only a part of the function’s domain.

So, that was about “domain stretching” to uncover hidden properties of a function. I was sorely tempted to discuss this beautiful concept here. So, I had to make some room for it.

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At this point, you are free to jump to Equation 2 below, from where our actual discussion of conditional and absolute convergence begins. Or, you can just stay on and see why Equation 2 is true.

On integrating Equation 1, we get:

-log(1 – x) = x + x²/2 + x³/3 + x⁴/4 + x⁵/5 + …

Therefore, log(1 – x) = – x – x²/2 – x³/3 – x⁴/4 – x⁵/5 – …

At x = -1, we get:

Equation 2: 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + … = log 2

Let us denote this series (the LHS of Equation 2) as S. So, S converges to log 2. However, for S to converge to log 2, there is a condition that needs to be satisfied: the terms have to be added in that order. If you add the terms in a different order, the series might either converge to a different quantity or diverge. For example, let us rearrange the terms in this series as follows:

1 – 1/2 – 1/4 + 1/3 – 1/6 – 1/8 + 1/5 – 1/10 – …

Or (1 – 1/2) – 1/4 + (1/3 – 1/6) – 1/8 + (1/5 – 1/10) – …

This is equivalent to 1/2 – 1/4 + 1/6 – 1/8 + 1/10 – …

This simplifies to 1/2 (1 – 1/2 + 1/3 – 1/4 + 1/5 – …) or 1/2 (S)

The rearranged series sums up to half of S!

Such series, whose limit depends on the order in which their terms are arranged, are said to be conditionally convergent. Those series that converge to the same quantity, no matter what order they are summed in, are said to be absolutely convergent.

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Other articles in this series: On Convergence, On Divergence