B4

Part of 1965 Putnam

Problems(1)

Source:

Consider the function

f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots},

n

is a positive integer. Express

f(x,n+1)

rationally in terms of

f(x,n)

x

. Hence, or otherwise, evaluate

\textstyle\lim_{n\to\infty}f(x,n)

for suitable fixed values of

x

\textstyle\binom nr

represent the binomial coefficients.)