 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/1/b/solution

coth x = \frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}} = \frac{e ^{2 x} + 1}{e ^{2 x} - 1}, x \neq = 0.

Expand the numerator and denominator using the exponential Maclaurin series:

coth x = \frac{1}{x} \frac{1 + x ^{2} /2 + x ^{4} /24 + O ( x ^{6} )}{1 + x ^{2} /6 + x ^{4} /120 + O ( x ^{6} )} = \frac{1}{x} + \frac{x}{3} - \frac{x ^{3}}{45} + O (x^{5}) .

This is a Laurent series, rather than an ordinary Maclaurin series:

coth x

has a simple pole at zero. The printed request should be understood as the expansion obtained from the exponential series, including this unavoidable

x^{- 1}

term.

 New to topics? Read the docs here!