Question

Determine whether each of the following is Always True, Sometimes True, or Always False. If the statement is Always True or Always False, provide a brief justification. If the statement is Sometimes True, provide an example of a series that makes it true and an example of a series that makes it false. In the following, {a_n}∞n=1 is a sequence and {s_n}∞n=1 refers to the corresponding sequence of partial sums.

(a) If lim n→∞ s_n = 0, then lim n→∞ a_n = 0.

(b) If lim n→∞ a_n = 0, then lim n→∞ s_n = 0.

(c) If lim n→∞ a_n = 0, then ∞∑k=1 a_k converges.

(d) If ∞∑k=1 a_k = 1, then lim k→∞ a_k = 1.

(e) If lim n→∞ a_n = 1, then lim n→∞ s_n = 1.

(f) If lim n→∞ s_n = 1, then ∞∑k=1 a_k converges.

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Answer 1

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