Look at that. It’s probably almost enough to cure you of any desire to learn maths. Or perhaps it makes you want to know what it means, in which case, you’ve come to the right place. With a bit of guidance, you can make things like this out of simple expressions.
How To Decode Expressions Like The One Above
First, to make this look a lot less scary, factorise the denominator. You’ll get:
That still looks as if it might give you nightmares, but it’s a little better.
(A brief note on factorisation:
Polynomials like the one in the original expression (ones with degree 2, meaning the maximum power is 2) have two solutions (a solution is the value of x for which f(x) = 0 (f(x) is just a way of expressing a function, such as y = 2x. Functions can be plotted on a graph and their solutions can be extracted from this graph)). They have two solutions because they can be simplified into two expressions of degree 1, as shown in the second picture. Once they’ve been reduced to the two degree-1 polynomials (a polynomial is an expression with more than one term), two solutions can be worked out using this. (For example, say f(x) = x2 + 5x + 6. It can be reduced to (x + 2)(x + 3). If x + 2 = 0, then you can find x; then the same idea can be applied to x + 3 = 0. Try putting the values obtained from this method into the original f(x), and check your answers!))
Back to the scary expression.
Note that (4x – 1) – (4x – 3) = 2, which just happens to be the numerator. So, you can make the original expression into:
That looks a lot simpler.
The sigma on the left means that k takes every value, in turn, from 1 to infinity, then adds all of the expressions together; basically, it starts with the value for k = 1, then adds the value for k = 2, then for k = 3 and so on to infinity.
So, if k = 1, what’s 4k – 3? It’s 1, and for 4k – 1, it’s 3. So, for k = 1, the expression means 1 – (1/3).
Continue substituting for k and adding, and you’ll get the value of the expression as:
which is the Leibniz approximation for π/4 – a comparatively simple answer compared to what we started with, which had a summation to infinity and a degree-2 polynomial.
Now it’s your turn to come up with complex and unnecessary expressions!