Give an example of each of the following, or state that such a request is impossible by referencing the proper theorem (s) : (a) sequences (xn) and (yn), which both converge, but whose sum (xn + yn) converges; (b) sequences (xn) and (yn), where (xn) converges, (yn) diverges, and (xn+yn) converges;

Question

Raymond Ward

Mathematics

18 October, 19:19

Give an example of each of the following, or state that such a request is impossible by referencing the proper theorem (s) : (a) sequences (xn) and (yn), which both converge, but whose sum (xn + yn) converges; (b) sequences (xn) and (yn), where (xn) converges, (yn) diverges, and (xn+yn) converges;

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Byron · Answer 1

There are many examples for the first request, but none for the second.

Step-by-step explanation:

a) There is a theorem which states that the sum of two convergent sequences is convergent, so any pair of convergent sequences (xn), (yn) will work (xn=1/n, yn=2/n, xn+yn=3/n. All of these converge to zero)

If you meant (xn) and (yn) to be both divergent, we can still find an example. Take (xn) = (n²) and (yn) = (1/n - n²). Then (xn) diverges to + ∞ (n² is not bounded above and it is increasing), (yn) diverges to - ∞ (1/n - n² is not bounded below, and this sequence is decreasing), but (xn+yn) = (1/n) converges to zero.

b) This is impossible. Suppose that (xn) converges and (xn+ýn) converges. Then (-xn) converges (scalar multiples of a convvergent sequence are convergent). Now, since sums of convergent sequences are convergent, (xn+yn + (-xn)) = (yn) is a convergent sequence. Therefore, (yn) is not divergent and the example does not exist.