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# 5.E: Convergence of the Taylor Series- A “Tayl” of Three Remainders (Exercises)

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## Q1

Find the Integral form, Lagrange form, and Cauchy form of the remainder for Taylor series for the following functions expanded about the given values of $$a$$.

1. $$f(x) = e^x$$, $$a = 0$$
2. $$f(x) = \sqrt{x}$$, $$a = 1$$
3. $$f(x) = (1 + x)^α$$, $$a = 0$$
4. $$f(x) = \frac{1}{x}$$, $$a = 3$$
5. $$f(x) = \ln x$$, $$a = 2$$
6. $$f(x) = \cos x$$, $$a = \frac{\pi }{2}$$

5.E: Convergence of the Taylor Series- A “Tayl” of Three Remainders (Exercises) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eugene Boman and Robert Rogers (OpenSUNY) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.