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babyagelesszj 2022-07-01 Answered
Let f ( x , y ) = n x x 2 + y n 2 . Show that g ( y ) = lim x f ( x , y ) exists for all y > 0. Find g ( y ).
My first impression of this problem is to use the monotone convergence theorem (MCT) or the dominated convergence theorem (DCT) to interchange the limit and the sum. However, I do not know what to bound the function by. Thanks in advance!
If we convert the sum to an integral under the counting measure μ on N , we can re-express the function as
f ( x , y ) = N x x 2 + y n 2 d μ
Ideally, we want to apply the DCT by finding an integrable function h such that | f n | = | x x 2 + y n 2 | h a.e. for all n. But, since h has to be an L 1 function independent of n under the counting measure, DCT might not be the correct method.
Also, the DCT that I learned involves interchanging lim n f n = lim n f n . But, this problem statement is asking for lim x and not lim n .
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Answers (1)

pompatzz8
Answered 2022-07-02 Author has 11 answers
We can calculate g ( y ) directly instead of interchanging limit and infinite sum:
g ( y ) = lim x n = 1 x x x 2 + y n 2 = lim x 1 x n = 1 x x 2 x 2 + y n 2 = lim x 1 x n = 1 x 1 1 + y ( n x ) 2 = 0 1 1 1 + y x 2 d x = 1 y 0 1 1 1 + ( y x ) 2 d y x = 1 y arctan y x | 0 1 = arctan y y .

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