Construct two divergent sequences (xn)n≥1 and (yn)n≥1 such that (xn + yn)n≥1converges
Question
Construct two divergent sequences and such that converges.
Solution
Sure, let's construct two divergent sequences (xn)n≥1 and (yn)n≥1 such that their sum (xn + yn)n≥1 converges.
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Define the sequence (xn)n≥1 as xn = n. This sequence is divergent because it tends to infinity as n increases.
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Define the sequence (yn)n≥1 as yn = -n. This sequence is also divergent because it tends to negative infinity as n increases.
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Now, let's consider the sequence (xn + yn)n≥1. For each n, we have xn + yn = n - n = 0.
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Therefore, the sequence (xn + yn)n≥1 is a constant sequence equal to 0, which is convergent.
So, we have constructed two divergent sequences (xn)n≥1 and (yn)n≥1 such that their sum (xn + yn)n≥1 converges.
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