Taylor and Maclaurin Series

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Taylor and Maclaurin Series

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Partial Derivatives

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Differentiability
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Double Integrals over General Regions

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Double integrals in polar coordinates

dA = r dr (d theta)
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Adding, Multiplying, and Dividing Power Series

Suppose that $\displaystyle f(x) = \sum_{n=0}^\infty a_n x^n$ and that $\displaystyle g(x) = \sum_{n=0}^\infty b_n x^n$. We can get the power series for $f(x)+g(x)$, $f(x)g(x)$ and $f(x)/g(x)$ by adding, multiplying, and dividing these expressions, as if they were polynomials:

$$f(x)+g(x) = \sum_{n=0}^\infty (a_n+b_n) x^n.$$
$$ f(x)g(x) = \sum_{n=0}^\infty c_n x^n, \hbox{ where } c_n = \sum_{i=0}^n a_i b_{n-i}.$$

Computing $f(x)/g(x)$, however, is trickier as we have to perform long division, and treat larger powers of $x$ as being less important than smaller powers. These ideas are explained in the following video. Unless you are told otherwise by your instructor, you will at the most be working with the sums and differences of series, and not the more complicated products and quotients.