Principles of Mathematics in Operations Research

186 13 Power Series and Special Functions

Remark 13.7.6 Let t = s^2 in the definition of T.

/>oo

r(x) = 2 s2x-le-a2 ds, 0 < x < oo.

Jo

The special case x — \ gives

/»oo

e~s ds — \Zn.

f

This yields

^yx— 1

^>^(1H^)

Remark 13.7.7 (Stirling's Formula) This provides a simple approximate

expression for T(x + 1) when x is large. The formula is

lim

F

^

+

S = 1.

*°° (f )XV2KX

Problems

13.1. Prove Theorem 13.1.8, the comparison tests for nonnegative series.

13.2. Discuss the convergence and divergence of the following series:

a) Eo°° £

b) ET f£>where« > o

c) E~(ei - 1)

•OSTMi + i)

e) Er 9 fc+V^> where V > °

13.3. One can model every combinatorial problem (instance r) as

^PXJ = r, Xi e St C Z+. Let A^- = s Q

Then, the power series

j&Si

0, j # Si

i j-0 k=0

is known as the generating function, where the number of distinct solutions to
^2t Xi = r is the coefficient ar. We know that, one can write down a generating
function for every combinatorial problem in such a way that ar is the number