Question

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

Short answer

The sequence $\left\{a_k\right\}$ is eventually decreasing sequence , if it is decreasing for some index $k$, where $k\in \mathrm{\mathbb{N}}$.

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 decreasing sequence, if it is decreasing for some index $k$, where $k\in \mathrm{\mathbb{N}}$.

Let $\left\{a_k\right\}$be the 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.

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$a_2\le a_1,a_3\le a_2,.....,a_{k+1}\le a_k$.