Chapter 7: Q. 12 (page 591)

Define what it means for a sequence $\{a_{k}\}$ to be eventually decreasing.

Short Answer

Expert verified

The sequence $\{a_{k}\}$ is eventually decreasing sequence , if it is decreasing for some index $k$ , where $k \in ℕ$ .

Step by step solution

Step 1. Given information

The given sequence is,

$\{a_{k}\}$ .

Step 2. Let ℕ>0.

The sequence $\{a_{k}\}$ is eventually decreasing sequence, if it is decreasing for some index $k$ , where $k \in ℕ$ .

Let $\{a_{k}\}$ be the sequence and $ℕ > 0$ be such that for all $k > ℕ, a_{k + 1} \leq a_{k,}$ then the sequence is eventually decreasing sequence.

Step 3. Consider the sequence n-n2n=1∞.

The terms of the sequence can be found by substituting $n = 1, 2, 3, . . .$

Hence, the terms of the sequence are, $1 - 1^{2}, 2 - 2^{2}, 3 - 3^{2}, . . . .$

The terms of the sequence are, $0, - 2, - 6, . . .$

The sequence is eventually decreasing sequence as $a_{2} \leq a_{1}, a_{3} \leq a_{2}, . . . . ., a_{k + 1} \leq a_{k}$ .

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Define what it means for a sequence $\{a_{k}\}$ to be eventually decreasing.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Let ℕ>0.

Step 3. Consider the sequence n-n2n=1∞.

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Define what it means for a sequence akto be eventually decreasing.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Let ℕ>0.

Step 3. Consider the sequence n-n2n=1∞.

One App. One Place for Learning.

Most popular questions from this chapter

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Define what it means for a sequence $\{a_{k}\}$ to be eventually decreasing.