Question

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

Short answer

The sequence$\left\{a_k\right\}$is eventually strictly decreasing sequence, if it is decreasing after 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 strictly decreasing sequence, if it is decreasing after 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 $k>\mathrm{\mathbb{N}},a_{k+1}<a_k$, then the sequence is eventually strictly 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 strictly decreasing sequence as $a_2<a_1,a_3<a_2,.....,a_{k+1}<a_k$.