Power series calculus functions can be used to approximate or replace complex functions with polynomials. Polynomials functions are well-behaved and easier to evaluate than other function types. By representing functions as infinite polynomials using sigma notation, power series allow for simpler calculations. For example, a power series can be used to approximate the sine function and adding more terms improves the accuracy of the approximation. Calculus teaches how to find and use these power series to simplify calculations.
There are many different types of functions in math: logarithmic functions, exponential functions, rational functions, and trigonometric functions, just to name a few. But these types of functions can be badly behaved, with things like vertical asymptotes and discontinuities, and difficult to evaluate.
Polynomials, on the other hand, are a type of function that is known for being well-behaved. A polynomial is an expression that consists of coefficients and nice (whole number) powers of a variable. For example, 4x + 7 is a polynomial of power 1 and x^2 + 3x – 1 is a polynomial of power 2. The thing to notice about these polynomials is that they don’t have asymptotes or discontinuities, and they are relatively easy to evaluate.
So, wouldn’t it be great if we could replace all - or at least pieces - of those badly-behaved functions with polynomials? Well, Calculus lets us do just that using what’s known as power series.
Think of a power series as a gentle giant polynomial. Instead of having a highest power of our variable (like a regular polynomial does), in a power series, the powers of our variable can go on forever and ever and ever…
But we can’t write out all the terms of this giant polynomial, right? So we use a shorthand notation.
We use the Greek symbol Sigma to mean “add or sum things up” and we use an index to keep track of the powers and coefficients of our giant polynomial. For example, our shorthand for x + ½ x^2 + 1/3x^3 and so on would be the sum from n = 1 to infinity of 1/n times x^n
In Calculus, we use power series like this to replace or approximate scary functions. For example, we can show that sin x equals the sum from n = 0 to infinity of (-1)^n over (2n+1)! times x^(2n+1) or, if we were to start writing out the terms x – x^3 over 6 + x^5/120 - + …
Then, we can do things like replace pieces of the graph of sin x using our gentle giant polynomial.
For instance, look at the graph of sin x compared with the graph of the first term of our series, which is just x.
See how the graphs are identical at x = 0 and hug each other for nearby values? This means that the function y=sin x is approximately equal to the polynomial y = x for values of x near 0. So, if we wanted to approximate the value of the sine function for some value close to 0, like .01, we could just put that value into the polynomial y = x and use that instead. Isn’t that much easier than evaluating the trigonometric function?
But notice how the graph of the first term of our power series and the graph of the sine function grow apart for values of x that are further away from zero? If we wanted the two graphs to be close together for a larger interval - so for more values of - we could just take more terms of our series.
For example, if we take the first two terms from our series, we can see that the graphs hug each other for even more values of x than when we just used the first term.
And we can keep doing this with however many terms in our power series we’d like to use. Calculus teaches us how to find these gentle giant polynomials and how they can make our lives, or at least our calculations, simpler.