Taylor Series - Math Is Fun

Taylor Series

A Taylor Series is an expansion of a function into an infinite sum of terms, where each term's exponent is larger and larger, like this:

Example: The Taylor Series for ex

ex = 1 + x + x22! + x33! + x44! + x55! + ...

says the function:ex is equal to the infinite sum of terms:1 + x + x2/2! + x3/3! + ... and so on

(Note: ! is the Factorial Function.)

Does it really work?

Let's try it for x = 2

Using a calculator we get e2 = (2.71828...)2 = 7.389056...

But let's try more and more terms of our infinite series:

Terms	Result
1+2	3
1+2+222!	5
1+2+222!+233!	6.3333...
1+2+222!+233!+244!	7
1+2+222!+233!+244!+255!	7.2666...
1+2+222!+233!+244!+255!+266!	7.3555...
1+2+222!+233!+244!+255!+266!+277!	7.3809...
1+2+222!+233!+244!+255!+266!+277!+288!	7.3873...

It starts out really badly, but it then gets better and better!

Try using "2^n/fact(n)" and n=0 to 20 in the Sigma Calculator and see what you get.

Here are some common Taylor Series:

Taylor Series expansion	As Sigma Notation
ex = 1 + x + x22! + x33! + ...	∞ Σ n=0 xnn!
sin x = x − x33! + x55! − ...	∞ Σ n=0 (−1)n(2n+1)!x(2n+1)
cos x = 1 − x22! + x44! − ...	∞ Σ n=0 (−1)n(2n)!x2n
11 − x = 1 + x + x2 + x3 + ... for |x| < 1	∞ Σ n=0 xn

(There are many more.)

Approximations

We can use the first few terms of a Taylor Series to get an approximate value for a function.

Here we show better and better approximations for cos(x):

1 − x2/2!
1 − x2/2! + x4/4!
1 − x2/2! + x4/4! − x6/6!
1 − x2/2! + x4/4! − x6/6! + x8/8!

Try it yourself here: images/function-graph.js?fn0=1-x%5E2%2F2%2Bx%5E4%2F24&fn1=cos(x)&xmin=-4.5&xmax=4.5&ymin=-2.6&ymax=2.6

You can also see the Taylor Series in action at Euler's Formula for Complex Numbers

What's this Magic?

How can we turn a function into a series of power terms like this?

Well, it isn't really magic. First we say we want to have this expansion:

f(x) = c0 + c1(x-a) + c2(x-a)2 + c3(x-a)3 + ...

Then we choose a value "a", and work out the values c0 , c1 , c2 , ... and so on

And it is done using derivatives (so we must know the derivative of our function)

Quick review: a derivative gives us the slope of a function at any point.

These basic derivative rules can help us:

• The derivative of a constant is 0
• The derivative of ax is a (example: the derivative of 2x is 2)
• The derivative of xn is nxn-1 (example: the derivative of x3 is 3x2)

We'll use the little mark ' to mean "derivative of".

OK, let's start:

To get c0, choose x=a so all the (x-a) terms become zero, leaving us with:

f(a) = c0

So c0 = f(a)

To get c1, take the derivative of f(x):

f'(x) = c1 + 2c2(x-a) + 3c3(x-a)2 + ...

With x=a all the (x-a) terms become zero:

f'(a) = c1

So c1 = f'(a)

To get c2, do the derivative again:

f''(x) = 2c2 + 3×2×c3(x-a) + ...

With x=a all the (x-a) terms become zero:

f''(a) = 2c2

So c2 = f''(a)/2

In fact, a pattern is emerging. Each term is

• the next higher derivative ...
• ... divided by all the exponents so far multiplied together (for which we can use factorial notation, for example 3! = 3×2×1)

And we get:

f(x) = f(a) + f'(a)1!(x-a) + f''(a)2!(x-a)2 + f'''(a)3!(x-a)3 + ...

Now we have a way of finding our own Taylor Series:

For each term: take the next derivative, divide by n!, multiply by (x-a)n

Example: Taylor Series for cos(x)

Start with:

f(x) = f(a) + f'(a)1!(x-a) + f''(a)2!(x-a)2 + f'''(a)3!(x-a)3 + ...

The derivative of cos is −sin, and the derivative of sin is cos, so:

• f(x) = cos(x)
• f'(x) = −sin(x)
• f''(x) = −cos(x)
• f'''(x) = sin(x)
• and so on...

And we get:

cos(x) = cos(a) − sin(a)1!(x-a) − cos(a)2!(x-a)2 + sin(a)3!(x-a)3 + ...

Now put a=0, which is nice because cos(0)=1 and sin(0)=0:

cos(x) = 1 − 01!(x-0) − 12!(x-0)2 + 03!(x-0)3 + 14!(x-0)4 + ...

Simplify:

cos(x) = 1 − x2/2! + x4/4! − ...

Try that for sin(x) yourself, it will help you to learn.

Or try it on another function of your choice.

The key thing is to know the derivatives of your function f(x).

Note: A Maclaurin Series is a Taylor Series where a=0, so all the examples we have been using so far can also be called Maclaurin Series.

8963, 8964, 8965, 8966, 8968, 8969, 8967, 8970, 8971, 8972 Algebra Index Copyright © 2026 Rod Pierce