Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) 1 / (1 + 2² + 3² + ⋯ + n²)

검증된 단계별 안내

First, recognize the general term of the series: \( a_n = \frac{1}{1^2 + 2^2 + 3^2 + \cdots + n^2} \).

Recall the formula for the sum of the first \( n \) squares: \( 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6} \). Substitute this into the term to get \( a_n = \frac{1}{\frac{n(n+1)(2n+1)}{6}} = \frac{6}{n(n+1)(2n+1)} \).

Analyze the behavior of \( a_n \) for large \( n \). Since \( n(n+1)(2n+1) \) is a cubic polynomial in \( n \), \( a_n \) behaves roughly like \( \frac{6}{2n^3} = \frac{3}{n^3} \) as \( n \to \infty \).

Compare the series \( \sum a_n \) to the p-series \( \sum \frac{1}{n^p} \) with \( p = 3 \). Since \( p > 1 \), the p-series converges.

By the Comparison Test, since \( a_n \) behaves like \( \frac{3}{n^3} \) and \( \sum \frac{1}{n^3} \) converges, the original series \( \sum_{n=1}^\infty \frac{1}{1^2 + 2^2 + \cdots + n^2} \) also converges.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges.

Sum of Squares Formula

The sum of the first n squares is given by the formula 1² + 2² + ⋯ + n² = n(n + 1)(2n + 1)/6. This formula helps simplify the denominator in the given series, making it easier to analyze the behavior of the terms as n becomes large.

Comparison Test for Series

The comparison test involves comparing the terms of a given series to those of a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than those of a divergent series, it diverges. This test helps determine convergence without finding the exact sum.

Determining Convergence or DivergenceWhich of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.∑ (from n=1 to ∞) 1 / (1 + 2² + 3² + ⋯ + n²)