Questions

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 59 (13.8 and 13.9)

Multiple choice

Let 𝑆 = ∑ 𝑛 = 0 ∞ 𝑎 𝑛 be a series. Which of the following statements must be true? Select all the correct answers.

Options

A.IF lim 𝑛 → ∞ 𝑎 𝑛 ≠ 0 , THEN the series S is DIVERGENT.

B.IF the series S is DIVERGENT, THEN lim 𝑛 → ∞ 𝑎 𝑛 = 0 .

C.IF lim 𝑛 → ∞ 𝑎 𝑛 = 0 , THEN the series S is CONVERGENT.

D.IF ∑ 𝑛 = 1 ∞ 𝑎 𝑛 is DIVERGENT, THEN ∑ 𝑛 = 1 ∞ 𝑎 𝑛 = ∞

E.IF ∑ 𝑛 = 1 ∞ 𝑎 𝑛 is DIVERGENT, THEN lim 𝑛 → ∞ 𝑎 𝑛 ≠ 0 .

F.IF the series S is CONVERGENT, THEN lim 𝑛 → ∞ 𝑎 𝑛 = 0 .

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Step-by-Step Analysis

We start by restating the problem: we have a series S = sum from n = 0 to ∞ of a_n, and we must determine which of the given statements must be true. Option 1: 'IF lim_{n→∞} a_n ≠ 0, THEN the series S is DIVERGENT.' - This is a standard necessary condition for convergence: if a_n does not tend to 0, the series cannot converge. Therefore, this statement must be true. Option 2: 'IF the series S is DIVERGENT, THE......Login to view full explanation

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