How do you show an infinite product converges?

To show an infinite product converges, you typically relate it to the convergence of an infinite series.

Convergence Criteria for Infinite Products

The most common approach for determining the convergence of an infinite product relies on its relationship to an infinite series. Specifically, consider an infinite product of the form:

∏_n=1^∞ (1 + a_n)

where a_n is a sequence of complex numbers.

Theorem:

If 0 ≤ a_n < 1 for all n, then the infinite product ∏_n=1^∞ (1 + a_n) converges if and only if the infinite series ∑_n=1^∞ a_n converges. Similarly, the infinite product ∏_n=1^∞ (1 - a_n) converges to a non-zero value if and only if the infinite series ∑_n=1^∞ a_n converges.

Explanation:

• Convergence & Series: This theorem provides a direct link between the convergence of the infinite product and the convergence of the series. If you can demonstrate that the series ∑ a_n converges (using tests like the comparison test, ratio test, or integral test), then you've also shown that the product ∏ (1 + a_n) converges. The same logic applies to ∏ (1 - a_n).

• Non-Zero Convergence: Note the emphasis on "converges to a non-zero value" in the case of ∏(1 - a_n). If ∑ a_n diverges, ∏(1 - a_n) could converge to zero.

Steps to Show Convergence:

1. Express the Infinite Product: Write the infinite product in the form ∏_n=1^∞ (1 + a_n) or ∏_n=1^∞ (1 - a_n).

2. Identify a_n: Determine the sequence a_n.

3. Check the Condition 0 ≤ a_n < 1: Verify that the condition 0 ≤ a_n < 1 holds (or that a_n is close to zero). If a_n is negative or greater than or equal to 1, this method doesn't directly apply.

4. Analyze the Series ∑ a_n: Investigate the convergence of the infinite series ∑_n=1^∞ a_n. Use any appropriate convergence test (e.g., comparison test, ratio test, root test, integral test).

5. Conclude:

   • If ∑ a_n converges, then ∏ (1 + a_n) converges.
   • If ∑ a_n diverges and 0 ≤ a_n < 1, then ∏ (1 + a_n) diverges.
   • If ∑ a_n converges, then ∏ (1 - a_n) converges to a non-zero value.
   • If ∑ a_n diverges, then ∏ (1 - a_n) may converge to zero or diverge. Further analysis is required in this case.

Example:

Consider the infinite product:

∏_n=1^∞ (1 + 1/n²)

Here, a_n = 1/n². Since 0 ≤ 1/n² < 1 for all n > 1, we can consider the series:

∑_n=1^∞ 1/n²

This is a p-series with p = 2, and it is known to converge (p-series test). Therefore, the infinite product ∏_n=1^∞ (1 + 1/n²) converges.

General Case

For a more general infinite product ∏(1 + z_n) where z_n are complex numbers, the product converges if and only if ∑ z_n converges and 1 + z_n ≠ 0 for all n. The convergence of ∑ |z_n|² is often useful in analyzing such cases.