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 Mission. 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: $$f(x)$$ = $$f(0) + \frac{dy}{dx}x+(\frac{dy}{dx})^2(0)*\frac{1}{2}*x^2$$

$$+(\frac{dy}{dx})^3(0)*\frac{1}{6}*x^3+(\frac{dy}{dx})^4(0)*{1}{24}*x^4...$$

Ratio = $$(\frac{dy}{dx})^n(0)*\frac{x^n}{n!}$$

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.