Explanation The limit comparison test states as follows: Let { a n } and { b n } be positive sequences. If the limit L = lim n → ∞ ( a n b n ) exists. Then (i) If L > 0 then ∑ a n converges if and only if ∑ b n converges. (ii) If L = ∞ and ∑ a n converges, then ∑ b n converges. (iii) If L = 0 and ∑ b n converges, then ∑ a n converges. Given: a n = 1 − 1 − 1 n Proof: Calculating the limit of a n = 1 − 1 − 1 n lim n → ∞ a n = lim n → ∞ ( 1 − 1

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Chapter 11, Problem 53CRE

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To prove:

The limit $\sum_{n = 1}^{\infty} a_{n} = 0$ and the series $\sum_{n = 1}^{\infty} a_{n}$ diverges.

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Chapter 11, Problem 52CRE

Chapter 11, Problem 54CRE

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To prove: The limit ∑ n = 1 ∞ a n = 0 and the series ∑ n = 1 ∞ a n diverges.

Solution Summary: The author explains how the limit comparison test proves that the series a n is also divergent.

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To prove:

The limit $\sum_{n = 1}^{\infty} a_{n} = 0$ and the series $\sum_{n = 1}^{\infty} a_{n}$ diverges.

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Chapter 11 Solutions

To prove: The limit ∑ n = 1 ∞ a n = 0 and the series ∑ n = 1 ∞ a n diverges.

Solution Summary: The author explains how the limit comparison test proves that the series a n is also divergent.

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To prove: The limit ∑n=1∞an=0 and the series ∑n=1∞an diverges.

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Chapter 11 Solutions

To prove:

The limit $\sum_{n = 1}^{\infty} a_{n} = 0$ and the series $\sum_{n = 1}^{\infty} a_{n}$ diverges.