Give an example of two divergent sequences (an) and (bn) such that their sum (an+bn) is convergent.

Give an example of two divergent sequences (an) and (bn) such that their sum (an+bn) is convergent.

Question
Sequences
asked 2021-02-09
Give an example of two divergent sequences (an) and (bn) such that their sum (an+bn) is convergent.

Answers (1)

2021-02-10
Consider the sequences an=(n^2)/(n-1) adn bn=(-n^2)/(n-1). Since liman=∞
n->∞ and limbn=-∞, the sequences are divergent. Their sum, however, is n->∞ an+bn=((n^2)/(n-1)-(n^2)/(n-1))=0. Therefore, lim(an+bn)=0 so the sum is n->∞ convergent.
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