Taylor series, the convergence of sequence, and series of functions
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Let Æ’n : R2 ï‚® R be Æ’n (x,y) = (x+y) / n
a. Show that Æ’n ï‚® 0 , but convergence is not uniform.
b. Show that the convergence Æ’n ï‚® 0 is uniform on bounded sets.
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1. Let Æ’n : R2 ï‚® R be Æ’n (x,y) = (x+y) / n
a. Show that Æ’n ï‚® 0 , but convergence is not uniform.
Proof. Given , for any we know that there exists when , we have
.
So, , i.e., Æ’n ï‚® 0.
But from the proof, we know that ...
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