Textbook Question
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)
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14. Sequences & Series
Series
1:52 minutesProblem 10.4.47d
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Every partial sum Sₙ of the series ∑ (k = 1 to ∞) 1 / k² underestimates the exact value of ∑ (k = 1 to ∞) 1 / k².
Verified step by step guidance1
Recall that the series \( \sum_{k=1}^\infty \frac{1}{k^2} \) is a convergent series with positive terms, known as the p-series with \( p = 2 > 1 \).
The partial sum \( S_n = \sum_{k=1}^n \frac{1}{k^2} \) represents the sum of the first \( n \) terms of the series.
Since all terms \( \frac{1}{k^2} > 0 \), the sequence of partial sums \( S_n \) is strictly increasing, meaning \( S_1 < S_2 < S_3 < \cdots \).
Because the series converges to a finite limit \( S = \sum_{k=1}^\infty \frac{1}{k^2} \), and the partial sums increase towards this limit, each partial sum \( S_n \) must be less than or equal to \( S \).
Therefore, every partial sum \( S_n \) underestimates the exact value of the infinite series \( \sum_{k=1}^\infty \frac{1}{k^2} \), making the statement true.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Sums of Infinite Series
A partial sum Sₙ is the sum of the first n terms of an infinite series. It approximates the total sum, and understanding how these sums behave helps determine if they overestimate or underestimate the series' exact value.
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Convergence and Monotonicity of Series
A series converges if its partial sums approach a finite limit. For series with positive, decreasing terms like 1/k², partial sums are increasing and approach the limit from below, which affects whether they underestimate or overestimate the total sum.
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Comparison and Counterexamples in Series Analysis
To verify statements about series, one uses comparison tests or constructs counterexamples. For the series ∑ 1/k², known results and inequalities help confirm if partial sums underestimate the total sum or not.
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