Please see the attached file for the fully formatted problems.
? Calculate the Taylor Polynomial and the Taylor residual for the function .
? Prove that as , for all .
? Find the Taylor series of f.
? What is the radius of convergence for the Taylor series? Justify your answer.
? Let f:[0,1] be a bounded function. Show that, for every n, the upper Riemann sum and the lower Riemann sum satisfy the inequality .
? Define for . Use upper and lower Riemann sums to prove that h is Riemann integrable.
? Use mathematical induction to prove that
? Let with . Calculate the upper Riemann sum and the lower Riemann sum of f on [0,1].
? Calculate the upper Riemann integral and the lower Riemann integral.
? Show that f is Riemann integrable in [0,1] and find the Riemann integral.
Please see the attached file for the full solution.
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1. We know , then , . In general, . Thus .
(a) By Taylor expansion, we have , where the Taylor polynomial is , the Taylor Residue is for some .
(b) We know for any fixed number , as . Now for any , we have . Thus as .
(c) The Taylor series of is ...
Riemann Sums, Taylor Polynomials, Taylor Residuals and Radius of Convergence are investigated. The solution is detailed and well presented.