Concrete mathematics : a foundation for computer science

9.6 FINAL SUMMATIONS 469
This is a doubly infinite sum, whose terms reach their maximum value e” = 1
when k = 0. We Cal:1 it 0, because it is a power series involving the quantity
eel/” raised to the p (k)th power, where p(k) is a polynomial of degree 2; such
power series are traditionally called “theta functions!’ If n = 1O1oo, we have
e- 01 M 0.99005, when k = 1049;
e k2/n - ec’ z 0.36788, when k = 105’;
e-lOO < 10P43, when k = 105’.
So the summand stays very near 1 until k gets up to about fi, when it
drops off and stays very near zero. We can guess that 0, will be proportional
to fi. Here is a graph of eekzin when n = 10:
Larger values of n just stretch the graph horizontally by a factor of $7.
We can estimate 0, by letting f(x) = eex2/” and taking a = -00, b =
$00 in Euler’s summation formula. (If infinities seem too scary, let a = -A
and b = +B, then take limits as A, B + 00.) The integral of f(x) is
if we replace x by u.fi. The value of s,” eeU2 du is well known, but we’ll
call it C for now and come back to it after we have finished plugging into
Euler’s summation formula.
The next thing we need to know is the sequence of derivatives f’(x),
f"(X), . . . . and for this purpose it’s convenient to set
f(x) = s(x/Jq , g(x) = epK2
Then the chain rule of calculus says that
df(x) Q(y) dy
- = - - -
dx dy dx ’
and this is the same as saying that
f'(x) = 5 g'(x/fi).
By induction we have
fck'(x) = nPk’2g(k1(x/fi).
y = -If_;
fi