cn x = 61 • • • bn

3=2

1 +

1

i - i

We all know from calculus that

j - i

J

lim 1 + -

e.

We therefore write

where

1 •?

1 + 1 -0.

Suppose we have numbers a3; — 0. When can we conclude that

the limit

n

iim

n ^ + ^ i

exists? (Assume, to avoid trivial cases, that CLJ ^ —1 for all j.) A

necessary condition is that CLJ — » 0. But this is not sufficient. Let us

assume that a,j — 0. When dealing with large products it is often

easier to take logarithms, since logarithms convert products to sums.

We know from properties of limits that

n n

In lim TT[1 + a A = lim In TT[1 + cu],

with each limit existing if and only if the other limit exists. Also,

n n

In J ] [l +

aj

] = ^ l n [ l + oi].