Q. 94

Question

Let $\{a_{k}\}$ be a sequence. Prove Theorem 7.6 (a) along with the following variations:

(a) Show that when $a_{k - 1} - a_{k}$ ≥ 0 for every k ≥ 1, the sequence is increasing.

(b) Show that when $a_{k - 1} - a_{k}$ > 0 for every k ≥ 1, the sequence is strictly increasing.

(d) Show that when $a_{k - 1} - a_{k}$ < 0 for every k ≥ 1, the sequence is strictly decreasing.

Step-by-Step Solution

Verified

Answer

Proved

1Step 1. Given

Consider the sequence $\{a_{k}\}$

2Part (a) Step 2. Explanation

$Since, a_{k + 1} - a_{k} \geq 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} \geq 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} \geq 0 + a_{k} a_{k + 1} \geq a_{k} for k \geq 1 Hence, be definition the sequence is i n creasing sequence .$

3Part(b) Step 3. Explanation

$Since, a_{k + 1} - a_{k} > 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} > 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} > 0 + a_{k} a_{k + 1} > a_{k} for k \geq 1 Hence, be definition the sequence is strictly i n creasing sequence .$

4Part(c) Step 4. Explanation

$Since, a_{k + 1} - a_{k} \leq 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} \leq 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} \leq 0 + a_{k} a_{k + 1} \leq a_{k} for k \geq 1 Hence, be definition the sequence is d e creasing sequence .$

5Part(d) Step 5. Explanation

$Since, a_{k + 1} - a_{k} < 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} < 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} < 0 + a_{k} a_{k + 1} < a_{k} for k \geq 1 Hence, be definition the sequence is s t r i c t l y d e creasing sequence .$

Q. 93

Q. 95

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