Another example of an infinite series that the student has encountered in previous courses is the power series. Examples of such series are provided by Taylor and Maclaurin series.
As we saw in the previous chapter, representing functions as power series was a fruitful strategy for mathematicans in the eighteenth century (as it still is). Differentiating and integrating power ser...As we saw in the previous chapter, representing functions as power series was a fruitful strategy for mathematicans in the eighteenth century (as it still is). Differentiating and integrating power series term by term was relatively easy, seemed to work, and led to many applications. Furthermore, power series representations for all of the elementary functions could be obtained if one was clever enough.
\[\begin{align} \int_{a}^{x}f^{(n-1)}(x)\cdot \Delta x-\int_{a}^{x}f^{(n-1)}(a) \cdot \Delta x &= \left[ f^{(n-2)}(x)-f^{(n-2)}(a) \right]-f^{(n-1)}(a)\int_{a}^{x} \Delta x \\ &\text{(Remembering $f^{...∫xaf(n−1)(x)⋅Δx−∫xaf(n−1)(a)⋅Δx=[f(n−2)(x)−f(n−2)(a)]−f(n−1)(a)∫xaΔx(Remembering f(n−1)(a) is a constant)=[f(n−2)(x)−f(n−2)(a)]−f(n−1)(a)(x−a).
