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What are the differences between Taylor and Maclaurin Series?

Taylor Series: Centered at any point 'a'. Maclaurin Series: Centered specifically at x=0.

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What are the differences between Taylor and Maclaurin Series?

Taylor Series: Centered at any point 'a'. Maclaurin Series: Centered specifically at x=0.

What are the differences between Taylor Series and Taylor Polynomials?

Taylor Series: Infinite sum of terms. Taylor Polynomials: Finite sum of terms (partial sum).

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.

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

What is the Maclaurin series for exe^x?

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

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)!}

What is the Maclaurin series for cos(x)?

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

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

∑n=0∞xn\sum_{n=0}^\infty x^n

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

∑n=0∞(−x)n\sum_{n=0}^\infty (-x)^n

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

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

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

∑n=0∞x2n\sum_{n=0}^\infty x^{2n}

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}

What is the Binomial Series?

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