MAT 137Y: Calculus with proofs
Assignment 8
Due on Sunday, Apr 6 by 11:59pm via GradeScope
1. Suppose the series converges at x = 3. What can we conclude about the convergence or the divergence of the following series? If both cases are possible, give two examples such that one is convergent and another one is divergent. If one case is true, explain why.
(a)
(b)
(c)
2. In this project, we estimate π much more effectively by using the arc tangent series and the relation
Note that converges too slowly for computational purposes.
(a) Calculate the Taylor series for arctan(x) centred at 0 and determine its radius of convergence.
(b) Show that
using the following addition formula:
if | arctan(x) + arctan(y)| < then arctan(x) + arctan(y) = arctan
Hint: what is tan(2 arctan ) ? what is tan(4 arctan ) ? How about tan(4 arctan −arctan )?
(c) Using the preceding two parts, find a power series formula for .
(d) Estimate π with a finite sum such the the error is within 10−6 by using part (c). Justify your answer.
3. After making bank at the hedge fund, Mik decides to pursue his true passion, solving complex math-ematical problems, and begins his PhD in Operations Research, forming a research group focused on signal processing. Mik invites the talented MAT137 TAs to join him in tackling important problems in faster computations and optimizating numerical methods.
During one of their first meetings, Alisa suggests an important question: “Many root-finding methods converge too slowly. Can we speed them up?” After some brainstorming, Katrina proposes a theorem showing that Newton’s Method achieves quadratic convergence by using Taylor series.