Problem sheet 8 March 2nd, 2004

MT290 Complex variable

Ex. 1 Give the radius of convergence for each of the following series:

a. ∑_{n=1}^{∞} \frac{z^{n}}{n^{3}} ; b. ∑_{n=0}^{∞} \frac{z^{n}}{5^{n}}; c. ∑_{n=0}^{∞} \frac{z^{n}}{n^{3} + 5^{n}}; d. ∑_{n=0}^{∞} z^{n}5^{−n^{2}}

Ex. 2 Find power series for the following functions about the points stated and give the radius of convergence for each of the series.

a. \frac{1}{2 − z} about z = 0; b. \frac{1}{2 − z} about z = 12; c. \frac{5}{(1 − z)(4 + z)} about z = 0.

d. e^{z} about z = i. e. \frac{1}{3 − z} about z = 4i.

Ex. 3 Determine the radius of convergence for:

a) ∑_{n=0}^{∞} \frac{z^{n}}{7^{n}} ,  b) ∑_{n=0}^{∞} \frac{z^{5n}}{7^{n}}  c) ∑_{n=0}^{∞} \frac{z^{bn}}{a^{n}}  d) ∑_{n=0}^{∞} \frac{z^{n}}{n^{r}}  e) ∑_{n=0}^{∞} \frac{z^{n}}{\binom{3n}{n}}.

where in c) a and b and in d) r are positive real constants. Recall that \binom{m}{n} = \frac{m!}{n!(m−n)!} (see also exercise on n! below).

Ex. 4 Determine the power series of sin z in two different ways:

a) Use the definition of sin in terms of the complex exp function.

b) Use \frac{d^{2} sin z}{dz^{2}} = − sin z, cos(0) = 1 and the fact that sin is an odd function.

c) Use the definition of sinh z in terms of the exp to find its power series. Compare with sin z and deduce that sin(iz) = i sinh z.

Can you simplify the above calculations by means of Taylor’s theorem?

Ex. 5 Stirling’s formula is: n! ∼ \sqrt{2πn} (\frac{n}{e})^{n} . Use your pocket calculator to convince yourself for n = 10, 20, 30, .... List some values of \frac{n!}{\sqrt{2πn}(\frac{n}{e})^{n}}

Use this to find an approximation for \binom{2n}{n} and \binom{3n}{n}. Try to prove a much weaker form of Stirling’s formula as follows: ln(n!) = ln 1 + ln 2 + … + ln n. Approximate the sum by an integral. Find a) a lower and b) an upper bound for n!. If these bounds are not too far apart you have a good approximation to n!.