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

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cheekabooy · Accepted Answer

Consider the sequences&amp;nbsp;an=n2n−1&amp;nbsp;adn&amp;nbsp;bn=−n2n−1. The sequences are divergent because&amp;nbsp;liman=∞n≥∞&amp;nbsp;and&amp;nbsp;limbn=−∞. Their sum&amp;nbsp;n≥∞&amp;nbsp;an+bn=(n2n−1−n2n−1)=0.lim(an+bn)=0, the sum is&amp;nbsp;n≥∞&amp;nbsp;convergent.

user_27qwe · Accepted Answer

Let&#039;s consider the sequences (an) and (bn) defined as:an=(−1)nbn=(−1)n+1In this case, (an) alternates between -1 and 1 for each term, while (bn) alternates between 1 and -1 for each term.Now, let&#039;s compute the sum of these sequences:an+bn=(−1)n+(−1)n+1Simplifying this expression, we get:an+bn=(−1)n−(−1)n=0As we can see, the sum of the two sequences is constantly 0 for every term, regardless of the value of n. Thus, the sequence (an+bn) is a constant sequence, and it converges to 0.Therefore, we have found an example of two divergent sequences (an) and (bn) such that their sum (an+bn) is convergent.

RizerMix · Accepted Answer

Let&#039;s define (an) as the sequence of positive integers:an=nAnd let&#039;s define (bn) as the sequence of negative reciprocals of the powers of 2:bn=−12nNow, let&#039;s find the sum (an+bn):an+bn=n+(−12n)As n approaches infinity, the term −1/2n approaches zero. Therefore, the sum (an+bn) converges to infinity:limn→∞(an+bn)=∞Hence, we have found an example where the sequences (an) and (bn) are divergent, but their sum (an+bn) is convergent.

karton · Accepted Answer

Step 1:Let&#039;s consider the following sequences:an=(−1)n and bn=(−1)n+1.Now, let&#039;s calculate the sum of the sequences:an+bn=(−1)n+(−1)n+1.Step 2:Simplifying this expression, we have:an+bn=(−1)n+(−1)n·(−1).an+bn=(−1)n+(−1)n·(−1).an+bn=(−1)n+(−1)n+1.an+bn=(−1)n−(−1)n.an+bn=0.Therefore, the sum of the sequences (an+bn) is a constant sequence equal to zero. Since a constant sequence is always convergent, the sum of (an+bn) is convergent.

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

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