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  2. AP Calculus
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What is the general formula for a Taylor Series?

∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n∑n=0∞​n!f(n)(a)​(x−a)n

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What is the general formula for a Taylor Series?

∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n∑n=0∞​n!f(n)(a)​(x−a)n

What is the Maclaurin series for exe^xex?

∑n=0∞xnn!\sum_{n=0}^\infty \frac{x^n}{n!}∑n=0∞​n!xn​

What is the Maclaurin series for sin(x)?

∑n=0∞(−1)nx2n+1(2n+1)!\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}∑n=0∞​(−1)n(2n+1)!x2n+1​

What is the Maclaurin series for cos(x)?

∑n=0∞(−1)nx2n(2n)!\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}∑n=0∞​(−1)n(2n)!x2n​

What is the Maclaurin series for 11−x\frac{1}{1-x}1−x1​?

∑n=0∞xn\sum_{n=0}^\infty x^n∑n=0∞​xn

What is the Maclaurin series for 11+x\frac{1}{1+x}1+x1​?

∑n=0∞(−x)n\sum_{n=0}^\infty (-x)^n∑n=0∞​(−x)n

What is the Maclaurin series for 11+x2\frac{1}{1+x^2}1+x21​?

∑n=0∞(−x)2n\sum_{n=0}^\infty (-x)^{2n}∑n=0∞​(−x)2n

What is the Maclaurin series for 11−x2\frac{1}{1-x^2}1−x21​?

∑n=0∞x2n\sum_{n=0}^\infty x^{2n}∑n=0∞​x2n

What is the Maclaurin series for ln(1+x)?

∑n=0∞(−1)nxn+1n+1\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}∑n=0∞​(−1)nn+1xn+1​

What is the Binomial Series?

∑n=0∞(an)xn\sum_{n=0}^\infty {a \choose n}x^n∑n=0∞​(na​)xn

Why are Maclaurin series important?

They provide a foundation for constructing Taylor series for other functions and are frequently encountered.

How are Taylor series and polynomial approximations related?

Taylor series use polynomial approximations to represent functions as infinite series.

What is the significance of the center 'a' in a Taylor series?

The Taylor series approximates the function best near the center 'a'.

What is the relationship between the Maclaurin series of sin(x), cos(x) and exe^xex?

The Maclaurin series for exe^xex contains all the terms from the Maclaurin series of sin(x) and cos(x).

How do you find the Taylor series for cos(3x) centered at x=0?

  1. Recall the Maclaurin series for cos(x). 2. Substitute 3x for x in the series. 3. Simplify the expression.

How do you find a Taylor series centered at x=a?

  1. Find several derivatives of f(x). 2. Evaluate the derivatives at x=a. 3. Identify a pattern. 4. Write the Taylor series using the formula.

How do you find the first four terms of a Taylor series?

  1. Find the general Taylor series. 2. Plug in n=0, 1, 2, and 3 into the series. 3. Simplify each term.