Question

Define what it means for a sequence $\left\{a_k\right\}$to be eventually monotonic.

Short answer

The sequence $\left\{a_k\right\}$is eventually monotonic sequence, if it either eventually increasing or eventually decreasing.

Step-by-step solution

Step 1. Given information

The given sequence is,$\left\{a_k\right\}$.

Step 2. Let ℕ>0.

The sequence $\left\{a_k\right\}$is eventually monotonic sequence, if it either eventually increasing or eventually decreasing.

Let $\left\{a_k\right\}$be a sequence and $\mathrm{\mathbb{N}}>0$be such that for all $k>\mathrm{\mathbb{N}},a_{k+1}\le a_k$, then the sequence is eventually decreasing sequence.

Let $\left\{a_k\right\}$be a sequence and $\mathrm{\mathbb{N}}>0$be such that for all $k>\mathrm{\mathbb{N}}$,$a_{k+1}\ge a_k$, then the sequence is eventually increasing sequence.

In both the cases the sequence$\left\{a_k\right\}$is eventually monotonic 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 role="math" localid="1649240464231" $a_2\le a_1,a_3\le a_2,.....,a_{k+1}\le a_k$.

Hence, the sequence is eventually monotonic.

Consider the sequence $\frac{1}{2},\frac{2}{3},\frac{3}{4},....,\frac{n}{n+1},..$

The sequence is eventually increasing sequence as, $a_2\ge a_1,a_3\ge a_2,.....,a_{k+1}\ge a_k$.

Hence, the sequence is eventually monotonic.