11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 0 to ∞)k² · 1.001⁻ᵏ

Verified step by step guidance

1. Identify the given series: \( \sum_{k=0}^{\infty} k^{2} \cdot 1.001^{-k} \). Notice that the general term is \( a_k = k^{2} \cdot (1.001)^{-k} \).
2. Rewrite the term to see the exponential part clearly: \( a_k = k^{2} \cdot \left( \frac{1}{1.001} \right)^{k} \). Since \( \frac{1}{1.001} < 1 \), this suggests the terms involve a polynomial factor times an exponential decay.
3. Consider applying the Ratio Test, which is useful for series involving factorials, exponentials, or powers. The Ratio Test states to compute \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
4. Calculate \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^2 \cdot (1.001)^{-(k+1)}}{k^2 \cdot (1.001)^{-k}} = \frac{(k+1)^2}{k^2} \cdot \frac{1}{1.001} \).
5. Evaluate the limit \( L = \lim_{k \to \infty} \frac{(k+1)^2}{k^2} \cdot \frac{1}{1.001} = \left( \lim_{k \to \infty} \frac{(k+1)^2}{k^2} \right) \cdot \frac{1}{1.001} = 1 \cdot \frac{1}{1.001} < 1 \). Since \( L < 1 \), the Ratio Test confirms that the series converges.

Key Concepts

Geometric Series and Its Convergence

A geometric series has terms of the form ar^k, where r is the common ratio. It converges if |r| < 1 and diverges otherwise. Understanding this helps analyze series with exponential terms like 1.001⁻ᵏ, which can be rewritten as (1/1.001)^k, a geometric term with ratio less than 1.

Comparison and Limit Comparison Tests

These tests compare a given series to a known benchmark series to determine convergence. The limit comparison test is useful when terms involve polynomials and exponentials, helping to confirm if the series behaves like a convergent geometric series despite polynomial factors like k².

Effect of Polynomial Factors on Series Convergence

Polynomial terms like k² grow slower than exponential decay terms. When combined with a geometric term with ratio less than 1, the exponential decay dominates, often ensuring convergence. Recognizing this balance is key to justifying convergence in series with mixed polynomial and exponential terms.