Definition 1: Convergence of an infinite product
An infinite product ∏ (n ∈ N) cn is said to converge to a limit L ≠ 0 if the partial products Pn approaches L as n approaches ∞ where:
Pn = ∏ (0 ≤ m ≤ n) cm
It also makes sense to define divergence for an infinite product.
Definition 2: Divergence of an infinite product
An infinite product ∏ (n ∈ N) cn is said to diverge if either the product does not converge (according to definition 1 above) or if it converges to 0.
So, if any of the factors of an infinite product is 0, then, that infinite product is said to diverge.
Lemma 1: if ∏ cn is convergent, then (lim n → ∞) cn = 1.
Proof:
(1) Assume that an infinite product ∏ cn is convergent.
(2) Let Pn be the partial product of this infinite product such that:
Pn = ∏ (0 ≤ m ≤ n) cm
(3) Then, for each cn of the series:
cn = Pn/(Pn-1)
(4) Since the product is convergent, there exists L such that as n goes to ∞, Pn → L and Pn-1 → L.
(5) It therefore follows that lim (n → ∞) cn = lim (n → ∞) Pn/Pn-1 = L/L = 1.
QED
Corollary 1.1: lim(1 + an)
if an = cn - 1, then:
∏(1 + an) is convergent → lim (n → ∞) an = 0.
Proof:
(1) an = cn - 1
(2) lim (n → ∞) an = lim(n → ∞) cn - 1 = 1 - 1 = 0 [This follows from Lemma 1 above]
QED
Lemma 2: Criteria for Convergence of ∑ an
For any N ≥ 0, if ∑ (n=N, ∞) an is finite, then ∑ (n=0, ∞) an is finite where ai is a real number.
Proof:
(1) Let {an} be a sequence such that each an is a real number.
(2) Assume that ∑ (n=N, ∞) an is finite
(3) Clearly, if N is any natural number, then ∑ (n=0, n less than N) an is finite.
(3) Since, ∑ (n=0, ∞) an = ∑ (n=0, n less than N) an + ∑ (n = N, ∞), it follows that:
∑ (n=0, ∞) an is finite and therefore convergent.
QED
Lemma 3: Criteria for Convergence of ∏(1 + an)
If an ≠ -1 for n ∈ N, then:
∏ (1 + an) converges if and only if ∑ log(1 + an) converges
Proof:
(1) Assume ∑ log(1 + an) converges to S
(2) Let Sn = ∑ (m ≤ n) log(1 + am)
(3) Then:
eSn = ∏ (m ≤ n) (1 + am) since:
elog(x) + log(y) = elog(xy) = xy [See here for review of the properties of Log]
(4) Since lim (n → ∞) Sn = S, this implies that:
lim (n → ∞) esn = eS
(5) Thus, we have shown that:
lim (n → ∞) (1 + an) = eS
(6) Assume that ∏(1 + an) converges
(7) Let ε be a real number greater than 0.
(8) Then, by Lemma 1 above (see here for review of mathematical limits), there exists N such that:
abs(Pn/Pn-1 - 1) is less than ε if m,n ≥ N.
This follows since lim (n → ∞) Pn = L which means:
lim (n → ∞) Pn/Pn-1 = L/L = 1
(9) Let Pn = ∏ (m ≤ N) (1 + an)
(10) If m,n ≥ N, then:
Log(Pn/PN) = Log(Pm/PN * Pn/Pm) = Log(Pm/PN) + Log(Pn/Pm) [See here for properties of Log]
(11) Let m = n-1
(12) Since cn = Pn/Pn-1 = Pn/Pm = an + 1 (see Lemma 1 above for details), we have:
Log(Pn/PN) = Log(Pm/PN) + Log(1 + an)
(13) Since m = n-1 and since we can apply this same argument to m-1, m-2, etc. we get:
Log(Pn/PN) = [Log(Pm-1) + Log(1 + am)] + Log(1 + an) = ∑ (N is less than i ≤ n) Log(1 + ai)
(14) Since lim (n → ∞) Pn = L, it follows that:
lim (n → ∞) Log(Pn/PN) = Log(L/PN)
(15) This shows that:
lim (n → ∞) ∑ (n greater than N) Log(1 + an) = Log(L/PN)
(16) Using Lemma 2 above gives us:
∑ (n ≥ 0) Log(1 + an) is convergent.
QED
Lemma 4: If abs(z) ≤ (1/2), then abs(log(1 + z)) ≤ 2*abs(z)
Proof:
(1) If abs(z) is less than 1, then Log(1 + z) is holomorphic. [Details to be added later, in the mean time see here]
(2) Since Log(1 + z) is holomorphic (see here for definition of holomorphic),
Log(1 + z) = z - z2/2 + z3/3 - ... [See Theorem, here]
(3) Using the Triangle Inequality for Complex Numbers (see Lemma 3, here), we have:
abs(Log(1 + z)) ≤ abs(z) + abs(z2/2) + abs(z3/3) + ...
which is less than
abs(z) + abs(z)2 + abs(z)3 + ...
(4) Now, abs(z) + abs(z)2 + abs(z)3 + ... = abs(z)/[1 - abs(z)] [See Lemma 1, here]
(5) Since abs(z) ≤ (1/2), it follows that:
1/[1 - abs(z)] ≤ 1/[1 - (1/2)] = 1/(1/2) = 2.
(6) So that we have:
abs(Log(1 + z)) ≤ 2*abs(z)
QED
Lemma 5: if ∑ an converges, there exists an natural number N such that for all n ≥ N, an is less than (1/2).
Proof:
(1) If ∑ ai converges, then only a finite number of ai ≥ (1/2).
(a) Assume that there is an infinite number of ai such that ai ≥ (1/2)
(b) If we build a subsequence (see Definition 6, here for definition of subsequence) of just those values of ai where ai ≥ (1/2)
(c) Since there are an infinite number of them, it follows that ∑ ai = ∞
(d) But this is impossible since ∑ (i=1, ∞) ai converges in step #1, so we reject our assumption at step #1a.
(2) If there are only a finite number of ai where ai ≥ (1/2), then we can let N-1 = the last such ai ≥ (1/2).
(3) Then, it follows that for all i ≥ N, ai is less than (1/2).
QED
Lemma 6: Criteria for Convergence using ∑ abs(an)
if an ≠ -1 for n ∈ N, then:
∑ abs(an) is convergent → ∏(1 + an) is convergent
Proof:
(1) Assume that ∑ abs(an) converges such that ∑ abs(an) = L
(2) Then there exists N such that n ≥ N → abs(an) is less than (1/2) [See Lemma 5 above]
(3) Using Lemma 4 above with an, it follows that:
abs(Log(1 + an)) ≤ 2*abs(an)
(4) So that we have:
∑ abs(Log(1 + an)) ≤ ∑ 2*abs(an) = 2*∑ abs(an) = 2*L.
(5) The conclusion follows from applying Lemma 3 above.
QED
References
- James M. Hyslop, Infinite Series, Dover Publications, 2006.
- Complex Logarithms, Wikipedia
3 comments :
can I ask you for a proof or explanation of why an infinite product is said to diverge when the limit = 0? Thanks
Hi Kazbich,
You can find Wikipedia's explanation here.
Feel free to let me know if you have additional questions from the Wikipedia article.
Cheers,
-Larry
Hi Larry, I have used both your post and the wikipedia article and I think I get the main points. I have two questions, however. In the second part of the proof of convergence criteria for Product (1 + an), you start with the fact that abs(Pn/Pn-1 - 1) is smaller than a no. e greater than zero for m,n greater than N. I am right in thinking that is necessary in order to ensure than the function log (Pn/Pn-1) exists for the interval [N is smaller than m is smaller than n]? Secondly, is this connected to the fact that the sequence of partial products is said to diverge if it converges to zero, since lim (n - infinity) log (Pn/PN) is undefined if Pn converges to zero? Thanks
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