5.18 Taylor and maclaurin series

Taylor and maclaurin series

Lessons

Notes:
Note * We can represent a function f(x)f(x) about x=ax=a as a Taylor Series. A Taylor Series is in the form

n=0f(n)(a)n!(xa)n\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n

where f(n)(a)f^{(n)}(a) is the nn'th derivative at x=ax=a. If a=0a=0, then we call it a Maclaurin Series. A Maclaurin Series is in the form:

n=0f(n)(a)n!xn\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}x^n

Here are some formulas that may be of use:

ex=n=0xnn!e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}

sin(x)=n=0(1)nx2n+1(2n+1)!sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!}

cos(x)=n=0(1)nx2n(2n)!cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}
  • 1.
    Maclaurin Series
    Find the Taylor or Maclaurin Series of the following functions without using the formulas:
  • 2.
    Using the Formula to Find the Maclaurin Series
    Use the formulas to find the Maclaurin Series for the following functions:
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Taylor and maclaurin series

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