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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 ProblemsEdit

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.

## StrategiesEdit

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.