cn x = 61 • • • bn
i - i
We all know from calculus that
j - i
lim 1 + -
We therefore write
1 + 1 -0.
Suppose we have numbers a3; — 0. When can we conclude that
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
In lim TT[1 + a A = lim In TT[1 + cu],
with each limit existing if and only if the other limit exists. Also,
In J ] [l +
] = ^ l n [ l + oi].