Inifinite Sums

We are moving on to the last symbol shortly, \(i\). Before we do so I just want to lay a little ground work with infinite sums.

You do not need to actually know about infinite sums to understand about imaginary numbers. So why are we bothering to look at them now? Two reasons. Firstly because the notation for these sums looks frightening, but isn't, and it crops up from time to time in maths texts. So it helps to know what we are looking at. Secondly, and more importantly, imaginary numbers sound a bit, well, imaginary, and actually they are very, very real and can be demonstrated using infinite sums - which do not actually involve any imaginary numbers at all.

So what the hell are we talking about? We all know what sums are - right? Colloquially they are arithmetical exercises, and also perhaps algebraic exercises. You "do your sums" if you are carrying out these types of exercises. In a more formal sense "sum" can be used as a synonym of "add". So if I "sum" two numbers I add them together. So, infinite sums are not homework that never, never, ends, but instead an addition operation that never, never ends. By way of example this is a sum:

\[1+2+3+4+5+6\]

and this is an infinite sum:

\[1+2+3+4+5+6+\ldots\]

The dots at the end just mean, and so on. Incidentally, these dots are called an ellipsis. The pattern is obvious - the numbers being added increase by one each time. So instead of writing out all the natural numbers (which would take literally for ever), you just stick the dots on the end. The result of the first sum is twenty one, the result of the second sum is infinity.

Even if we are just writing out the first sum above, it does take up quite a bit of space. We can use a notation to represent this sum in much less space, albeit it looks quite scary to start with. So, what we want to do is firstly work out how we describe each individual number to be added. We intuitively know that the numbers above are a pattern. There is an obvious logic behind the numbers. Each one is one larger than the last one. A list of numbers like this, with a logic behind how you get the next one, is called a series of numbers. So what we want to do is work out how to get any particular number in the series if we are just told which position it has in the series. How do we show that? We use functions! Remember a function just takes a variable (like the position in the series) and then does things to it to produce a result.

Once we have done that, we need to show that we are adding lots of things together. We need to have some sort of symbol to represent lots of addition. We also need to show which position in the series we are going to start counting from, and when to stop. That should do it.

In fact the symbol used in maths is this:

\[\sum\]

All that says is that what follows it is going to be added up. We put our formula for working out the numbers to the right of the symbol like this:

\[\sum f(x)\]

We then show what position we start in the series below the symbol:

\[\sum_{x=1} f(x)\]

Which for us in our example above is the first position, so we start with \(x\) equal to one. Remember, the one just means the first position in the series which goes into the function, not the outcome of the function. Lastly we need to show at which position in the series we want to stop adding entries. We do this by putting that number at the top of the symbol:

\[\sum_{x=1}^6 f(x)\]

So that complicated mess of stuff just means take a series where you generate each entry by putting the position of the entry in the series into the function, put the numbers one to six into the function in turn, and add up all the outcomes. To generate the series above the function \(f(x)\) is just \(x\) because the output of the function is the same as the input.

To symbolise adding up the infinite series illustrated above, you would put infinity at the top of the summation symbol to show that you just kept adding and adding forever:

\[\sum_{x=1}^\infty f(x)\]

What if you wanted to add up all the even numbers? Remember in our proof of the irrationality of \(\sqrt{2}\) we said that any even number was a number that could be divided by two to give another whole number. So \(f(x)\) would be \(2\cdot x\). That would make the first entry in the series two multiplied by one, the second two multiplied by two and the third two multiplied by three, or 2, 4, 6 etc etc ellipsis.

It is traditional that instead of \(x\) the variable we use to denote the position in the sequence is \(n\). Typically we would also dispense with the \(f(n)\) stuff and we would write \(a_n\) instead. That's a bit bloody weird though, using two variables for one number. All that means is that \(a\) is the \(n^{th}\) number in the series. I prefer to stick with \(f(x)\) or at least set out the function, because that way you can be sure what is generating the series of numbers.