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].
Previous Page Next Page