Prove that from the convergence of product from n=1
Prove that from the convergence of $\displaystyle\prod_{n=1}^{\infty}p_n$ and $\displaystyle\prod_{n=1}^{\infty}q_n$ (to positive finite limits) it follows that $\displaystyle\prod_{n=1}^{\infty}p_nq_n$ and $\displaystyle\prod_{n=1}^{\infty}p_n/q_n$ also converge.
Explanation of Method
Partial products multiply and divide exactly like ordinary numbers: $\prod_{n=1}^N p_nq_n=P_NQ_N$ and $\prod_{n=1}^N p_n/q_n=P_N/Q_N$, so taking $N\to\infty$ and using the algebra of limits (product and quotient of convergent sequences, with the quotient's denominator limit nonzero) gives the result immediately — either directly, or equivalently via $\ln(p_nq_n)=\ln p_n+\ln q_n$ additivity of the logarithm.
- Relate partial products of $p_nq_n$ (resp. $p_n/q_n$) to $P_NQ_N$ (resp. $P_N/Q_N$).
- Apply the limit laws for products and quotients of convergent sequences.
- Cross-check via the additive logarithmic form.
Step 1 · Partial products factor exactly
With $P_N=\prod_{n=1}^Np_n\to P>0$ and $Q_N=\prod_{n=1}^Nq_n\to Q>0$: $$ \prod_{n=1}^N p_nq_n = P_NQ_N, \qquad \prod_{n=1}^N\frac{p_n}{q_n}=\frac{P_N}{Q_N}. $$
Step 2 · Take the limit
Since limits of products/quotients of convergent sequences equal the product/quotient of the limits (valid here since $Q\ne0$): $$ \boxed{\prod_{n=1}^{\infty}p_nq_n = \lim_{N\to\infty}P_NQ_N = PQ}, \qquad \boxed{\prod_{n=1}^{\infty}\frac{p_n}{q_n} = \lim_{N\to\infty}\frac{P_N}{Q_N}=\frac PQ.} $$
Step 3 · Logarithmic cross-check
Equivalently, $\sum\ln p_n=\ln P<\infty$ and $\sum\ln q_n=\ln Q<\infty$ (both convergent), so by linearity $$ \sum\ln(p_nq_n)=\sum\ln p_n+\sum\ln q_n=\ln P+\ln Q=\ln(PQ), \qquad \sum\ln\frac{p_n}{q_n}=\ln P-\ln Q=\ln\frac PQ, $$ recovering the same two results after exponentiating.
Summary Table
| Combination | Limit |
|---|---|
| $\prod p_nq_n$ | $PQ$ |
| $\prod p_n/q_n$ | $P/Q$ |
Additional Notes
- The requirement that both limits be positive and finite is essential: if either product converges to $0$ or diverges, the product/quotient rule can fail or become indeterminate (e.g. $\infty\cdot0$), exactly as with ordinary sequence limits.
- This algebraic closure property is what justified, throughout Problems 371–383, freely splitting a complicated product into simpler pieces (e.g. Problem 374's ratio of two Weierstrass $\sinh$ products, or Problem 372's factorization into two telescoping chains) and recombining the results.
- The logarithmic proof in Step 3 is really just Problem 384's underlying machinery (product $\leftrightarrow$ series via $\ln$) applied one level up: once convergence of a product is known, its logarithm is an ordinary convergent series, and all the usual linearity properties of series apply directly.