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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. $$f(x) = \cos x$$, $$a = \frac{\pi }{2}$$

### Contributor

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)