MathDB

Putnam 1965 B4

Source:

September 28, 2020

Putnam

Problem Statement

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},

where

n

is a positive integer. Express

f(x,n+1)

rationally in terms of

f(x,n)

and

x

. Hence, or otherwise, evaluate

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

for suitable fixed values of

x

. (The symbols

\textstyle\binom nr

represent the binomial coefficients.)

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