Math 1320 Lab EC
Name:
Infinite Products
Consider the polynomial function p(x) = x − x3 . There is another way to view this
polynomial, by factoring it as a product:
p(x) = x − x3 = x(1 − x2 ) = x(1 − x)(1 + x)
Both expressions are useful: x − x3 is easier to integrate and differentiate, while it is easier
to see that x(1 − x)(1 + x) is zero exactly when x is 0, 1, or −1.
You have learned this semester about how to express functions like sin(πx) as a power
series, a sort of “infinite polynomial” which, convergence aside, has the same kind of behavior
as a polynomial with respect to differentiation and integration.
sin(πx) =
∞
X
(−1)j (πx)2j+1
j=0
(2j + 1)!
= πx −
(πx)3 (πx)5
+
− ···
3!
5!
It turns out that, just like for polynomials, we can write down a product formula for sin(πx)
in terms of its zeroes.
x2
x2
x2
2
sin(πx) = πx 1 − x
1−
··· 1 − 2 ···
1−
4
9
N
Using both the infinite product formula and the Taylor series formula for sin(πx) allows us
to compute some interesting values of series, which is the purpose of this assignment.
1. This problem is about convincing yourself that
P the product formula actually represents
sin(πx). The test for divergence of a series ∞
n=1 an states that an must converge to 0
for the series to converge. Similarly, for an infinite product a1 a2 · · · an · · · to converge to
a nonzero real number, it is necessary for an to converge to 1.
(a) Show that the infinite product for sin(πx) does not fail the test for divergence for a
product. To do this, fix x to be some real number, and then compute the following
limit.
x2
lim 1 − 2 = 1
n→∞
n
Explicitly state any theorems about convergence of a sequence that you use.
(b) Find all real numbers x for which sin(πx) = 0. Show that these values make the
infinite product zero as well.
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2. Now we can start computing. Recall the product from the first page,
x2
x2
x2
2
1−
··· 1 − 2 ···
sin(πx) = πx 1 − x
1−
4
9
N
We can take the product of finitely many terms of the above expression to help us
understand the infinite product.
(a) Consider the finite product up to N = 2 and N = 3. What are the coefficients of
x3 in these polynomials?
(b) Based on your answer to the previous part, find a formula for the coefficient of x3 if
we compute the finite product for an arbitrary N ≥ 1. Take the limit as N → ∞,
which will give you an infinite series.
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3. Recall the power series for sin(πx).
sin(πx) = πx −
(πx)3 (πx)5
+
− ···
3!
5!
We can view the infinite product as a factorization of the above “infinite polynomial”.
Consider the coefficient of x3 in the infinite product for sin(πx), which you found in part
(b) of the last problem. This must be equal to the coefficient of x3 in the power series
for sin(πx). Use this equality to find a value for the following series.
∞
X
1
S=
n2
n=1
Page 4