Math Problem Statement

Evaluate the limit of the product as n approaches infinity: (1 - 1/2)(1 - 1/4) ... (1 - 1/2n).

Solution

The equation in the image is:

lim⁡n→∞(1−12)(1−14)⋯(1−12n)\lim_{n \to \infty} \left( 1 - \frac{1}{2} \right) \left( 1 - \frac{1}{4} \right) \cdots \left( 1 - \frac{1}{2n} \right)

This represents the limit of a product as nn approaches infinity. The product inside the limit is over terms of the form 1−12k1 - \frac{1}{2k} where kk ranges from 1 to nn.

Would you like a detailed solution or have any specific questions about this limit?

Here are some related questions:

  1. How do you evaluate the limit of a product as nn approaches infinity?
  2. What is the connection between this type of product and infinite series?
  3. How can you approximate this product for large nn?
  4. What mathematical tools are typically used to analyze such products?
  5. How does this expression relate to known infinite product results?

Tip: When dealing with limits that involve products or sums, it’s often useful to take the logarithm to simplify the expression before analyzing the limit.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Infinite Products

Formulas

lim_{n \to \infty} \prod_{k=1}^{n} \left( 1 - \frac{1}{2k} \right)

Theorems

Infinite Product Theorem
Limit Theorem

Suitable Grade Level

Undergraduate Level

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Limit of the Product Expansion: $$ \lim_{x \to 0} \left( \frac{1 - \prod_{k=1}^{n} (1 + x)}{x} \right) $$

Calculating Infinite Products: \( \prod_{k=1}^{\infty} \frac{3k - 2}{3k + 1} \), \( \prod_{k=1}^{\infty} \frac{4k + 2}{4k - 2} \), and \( \prod_{k=1}^{\infty} (3k - 2)(3k + 4) \)