Consider the function g(t)=f(x)−n∑k=0f(k)(t)k!(x−t)k−λ(n+1)!(x−t)n+1. Then \[g(\bar{x})=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(\bar{x})}{k !}(x-\bar{x})^{k...Consider the function g(t)=f(x)−n∑k=0f(k)(t)k!(x−t)k−λ(n+1)!(x−t)n+1. Then g(ˉx)=f(x)−n∑k=0f(k)(ˉx)k!(x−ˉx)k−λ(n+1)!(x−ˉx)n+1=f(x)−Pn(x)−λ(n+1)!(x−ˉx)n+1=0. and g(x)=f(x)−n∑k=0f(k)(x)k!(x−x)k−λ(n+1)!(x−x)n+1=f(x)−f(x)=0. By Rolle's theorem, there exists c in between ˉx and x such that…
ne of the most important uses of infinite series is the potential for using an initial portion of the series for f to approximate ff . We have seen, for example, that when we add up the first n terms ...ne of the most important uses of infinite series is the potential for using an initial portion of the series for f to approximate ff . We have seen, for example, that when we add up the first n terms of an alternating series with decreasing terms that the difference between this and the true value is at most the size of the next term. A similar result is true of many Taylor series.
