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

Short Answer

Expert verified

The sequenceakis eventually strictly decreasing sequence, if it is decreasing after some indexk, wherek.

Step by step solution

01

Step 1. Given information

The given sequence is,ak.

02

Step 2. Let ℕ>0.

The sequence akis eventually strictly decreasing sequence, if it is decreasing after some index k, where k.

Let akbe the sequence and >0be such that for all k>,ak+1<ak, then the sequence is eventually strictly decreasing sequence.

03

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, 1-12,2-22,3-32,......

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

The sequence is eventually strictly decreasing sequence as a2<a1,a3<a2,.....,ak+1<ak.

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