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Problem 1137

Test the series: \(1+2 ! / 2^{2}+3 ! / 3^{3}+4 ! / 4^{4}+\ldots \ldots\) by means of the ratio test. If this test fails, use another test.

Expert verified

Both the ratio test and the root test were inconclusive for determining the convergence or divergence of the given series \(\sum_{n=1}^{\infty} \frac{n!}{n^n}\). Further analysis using another test or comparison with known convergent or divergent series would be needed to establish convergence or divergence for this series.

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Chapter 36

Establish the convergence or divergence of the series: $\sin \pi / 2+1 / 4 \sin \pi / 4+1 / 9 \sin \pi / 6+1 / 16 \sin \pi / 8+\ldots .$

Chapter 36

Establish the convergence or divergence of the series: $[1 /(1+\sqrt{1})]+[1 /(1+\sqrt{2})]+[1 /(1+\sqrt{3})]+[1 /(1+\sqrt{4})]+\ldots$

Chapter 36

Test the series: $\left[1-3^{2} / 2^{2}\right]+\left[3^{4} /\left(2^{2} \cdot 4^{2}\right)\right]-\left[3^{6} /\left(2^{2} \cdot 4^{2} \cdot 6^{2}\right)\right]+\ldots \ldots \ldots$ by means of the ratio test. If this test fails, use another test.

Chapter 36

Determine the general term of the sequence: $$ 1 / 5^{3}, 3 / 5^{5}, 5 / 5^{7}, 7 / 5^{9}, 9 / 5^{11} $$

Chapter 36

Determine the general term of the sequence: $$ 1 / 2,1 / 12,1 / 30,1 / 56,1 / 90, \ldots $$

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