Derivation of the Taylor Series

Consider a function which can be represented as an infinite sum of monomial terms, like so:

click to view image in full size

Now, I will evaluate the function at x = a. Notice that all terms other than the first one will simplify to zero. So,

Now, I will differentiate the function:

I will evaluate the derivative function at x = a, just like I did with the original function. Again, it can be seen that all values except the first term will simplify to zero. So,

I am going to differentiate the function again:

Just like I did before, I am going to evaluate this function at x = a, and unsurprisingly, all terms except for the first term will simplify to zero. So,

And therefore:

Differentiating the function again will give me:

Evaluating this function at x = a, will give me:

And so, 

Differentiating the function yet again, I will get:

It should be obvious that the terms after the first would all simplify to zero if we evaluate this function at x = a. So,





and,







Let me list the coefficients again so that we may find a pattern for all other coefficients.
























A pattern should be clear to see now. First of all notice the numbers in the denominators. It is obvious that they are the factorials of the first few integers, although this may be less obvious for the first two coefficients, but this is only because that zero factorial and one factorial are both one. It seems that the nth coefficient is equal to the nth derivative evaluated at a, divided by n factorial In other words:

And indeed, this is the case. So, from knowing the above, the formula for the Taylor series is: