Let a: Z R be a function. We write an instead of a(n). We define the...

 

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Let a: Z R be a function. We write an instead of a(n). We define the limit 00 k n=- N n=-N n=k+1 where k E Z is some integer, if both limits on the right-hand side exist and the sum makes sense (i.e., they are either both finite, or both improper with the same sign). 2. 1. (3 points) Show that this definition is independent of the number k chosen. (3 points) Let (SN,M) be the double sequence SM,N = n=-Man. Show that -- an exists and is finite if and only if the double sequence (SM,N) converges, and that in this case the two limits coincide. an := lim N an + lim N = an (Hint: Write SN,M = AN + BM for sequences AN, BM. Show that SN,M converges if and only if both AN, BM converge. The Cauchy Criterion may help for one direction.) Let a: Z R be a function. We write an instead of a(n). We define the limit 00 k n=- N n=-N n=k+1 where k E Z is some integer, if both limits on the right-hand side exist and the sum makes sense (i.e., they are either both finite, or both improper with the same sign). 2. 1. (3 points) Show that this definition is independent of the number k chosen. (3 points) Let (SN,M) be the double sequence SM,N = n=-Man. Show that -- an exists and is finite if and only if the double sequence (SM,N) converges, and that in this case the two limits coincide. an := lim N an + lim N = an (Hint: Write SN,M = AN + BM for sequences AN, BM. Show that SN,M converges if and only if both AN, BM converge. The Cauchy Criterion may help for one direction.)

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1 To show that the definition is independent of the number k chosen we need to demonstrate that for any k and k if both limits on the righthand side e... View the full answer