If you're seeing this message, it means we're having trouble loading external resources for Khan Academy.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Maclaurin series and Euler's identity

8 videos
2 skills
In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.

Maclaurin and Taylor series intuition

VIDEO 12:59 minutes
Approximating a function at 0 using a polynomial

Cosine Taylor series at 0 (Maclaurin)

VIDEO 5:37 minutes
Approximating f(x)=cos x using a Maclauren Series (special case of a Taylor series at x=0)

Sine Taylor series at 0 (Maclaurin)

VIDEO 6:33 minutes
Sine Taylor Series at 0 (Maclaurin)

Taylor series at 0 (Maclaurin) for e to the x

VIDEO 6:10 minutes
Taylor Series at 0 (Maclaurin) for e to the x

Euler's formula and Euler's identity

VIDEO 11:27 minutes
Rationale for Euler's Formula and Euler's Identity

Integration and differentiation of power series

Integration and differentiation of power series