Maclaurin series for sin x, cos x, and e^x

The Maclaurin series for sin x, cos x, and e^x exercise appears under the Integral Calculus math section on Khan Academy. This exercise shows you how to turn a function into a power series.
==Types of Problems==
There are five types of problems in this exercise:
1. Determine the first three non-zero terms of the Maclaurin polynomial: The student is asked to find the first three non-zero terms of the Maclaurin polynomial for the given function.
2. Determine the sum of the infinite series given: The student is asked to find the exact value of the sum of the infinite series given.
3. Determine the value of the power series at the given point: The student is asked to evaluate the power series at a given point.
4. Determine what function evaluates to the given power series: The 
student is asked to match up the function that evaluates to the given power series. 
5. Determine the value of the point given the function: The student is asked to find out the value of point using the Maclaurin series on the function.
==Strategies==
Knowledge of taking derivatives, taking integrals, power series, and Maclaurin series are encouraged to ensure success on this exercise.
1. Maclaurin series: f(x) = f(0) + f	\prime(0)+	f\prime	\prime(0)*\frac{1}{2}*x^2
+	f\prime	\prime	\prime(0)*\frac{1}{6}*x^3+		f\prime	\prime	\prime\prime(0)*{1}{24}*x^4...
Ratio = 	f^n(0)*\frac{x^n}{n!}
2. The Maclaurin series is a special case of the Taylor series.
3. The Maclaurin series of sine is:
f(x) = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}...
4. The Maclaurin series of cosine is:
f(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}...
5. The Maclaurin series of e^x is:
f(x) = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}...
6. Euler's formula:
e^{ix} = cos(x)+isin(x)
7. Euler's identity:
e^{i\pi}+1=0
8. Complex functions can be converted to power series by using substitution.
Category:Math