PHYS2124 Mathematical Methods for Physics I
Homework 1
Due: Feb 27, 11:59pm on Canvas
Problem 1 (Ch 1. 16.8) (SQ) Test for convergence
∞
X
2n3
n4 − 2
n=2
Problem 2 (Ch 1. 16.12) (SQ) Find the interval of convergence, including the endpoint test for
∞
X
xn n2
5n (n2 + 1)
n=1
Problem 3 (Ch 1. 16.21) (SQ) Find the first 3 terms of the Taylor series for
ex about a = 1
Problem 4 (Ch 2. 17.3 & 11) (SQ) Find one or more values for
√
−1
(a) 5 −4 − 4i, (b)e2 tanh i
Problem 5 (Ch 2. 17.13) (SQ) Find real x and y for which |z + 3| = 1 − iz, where
z = x + iy.
Problem 6 (Ch 2. 17.21) (SQ) Verify the equation cosh−1 z = ln z ±
Problem 7 (Ch 2. 17.26) (SQ) Find
2eiθ −i
ieiθ +2
Problem 8 (Ch 2. 17.32) (SQ) Use a series you know to show that
∞
X
(1 + iπ)n
n=0
n!
1
= −e
√
z2 − 1
Optional Problem:[Ch 1. 16.1] Show that it is possible to stack a pile of identical
books so that the top book is as far as you like to the right of the bottom book. Start at
the top and each time place the pile already completed on top of another book so that the
pile is just at the point of tipping. (In practice, of course, you can’t let them overhang
quite this much without having the stack topple. Try it with a deck of cards.) Find the
distance from the right-hand end of each book to the right-hand end of the one beneath
it. To find a general formula for this distance, consider the three forces acting on book
n, and write the equation for the torque about its right-hand end. Show that the sum of
these setbacks is a divergent series (proportional to the harmonic series).
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