1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))

검증된 단계별 안내

Examine the general term of the series: \(a_k = \frac{k^2}{k^4 + k^3 + 1}\).

To understand the behavior of \(a_k\) for large \(k\), simplify the expression by dividing numerator and denominator by \(k^4\), the highest power in the denominator: \(a_k = \frac{\frac{k^2}{k^4}}{\frac{k^4}{k^4} + \frac{k^3}{k^4} + \frac{1}{k^4}} = \frac{\frac{1}{k^2}}{1 + \frac{1}{k} + \frac{1}{k^4}}\).

As \(k\) approaches infinity, the terms \(\frac{1}{k}\) and \(\frac{1}{k^4}\) approach zero, so \(a_k\) behaves like \(\frac{1/k^2}{1} = \frac{1}{k^2}\).

Since the series behaves like \(\sum \frac{1}{k^2}\) for large \(k\), which is a \(p\)-series with \(p=2 > 1\), the Comparison Test or Limit Comparison Test is appropriate to determine convergence.

Therefore, identify the Comparison Test or Limit Comparison Test as the convergence test to apply, comparing the given series to the convergent \(p\)-series \(\sum \frac{1}{k^2}\).

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주요 개념

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Comparison Test

The Comparison Test helps determine convergence by comparing a given series to a second series whose convergence behavior is known. If the terms of the original series are smaller than those of a convergent series, it also converges. Simplifying the terms to a comparable form is often necessary before applying this test.

Limit Comparison Test

The Limit Comparison Test involves taking the limit of the ratio of the terms of two series. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when the series terms are complicated but resemble a simpler series asymptotically.

Simplifying the General Term

Before applying convergence tests, it is important to simplify or rewrite the general term of the series to identify dominant terms. For example, factoring out the highest powers in numerator and denominator helps reveal the behavior of terms for large indices, making it easier to choose an appropriate test.

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))