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Mathematical Analysis

Section 7.2 Series of Functions

Definition 7.16.

Let (fn)n=1βˆžβŠ†B(X) for a set X. We'll say βˆ‘n=1∞fn converges uniformly (pointwise) if (βˆ‘n=1Nfn)N=1βˆžβŠ†B(X) converges uniformly (pointwise).

Remark 7.18.

This is a slightly non-standard version of the theorem. Combining this version with the comparison test gives the usual statement: If (fn)n=1βˆžβŠ†B(X) and (Mn)n=1βˆžβŠ†[0,∞) so that
  1. for all x∈R,nβ‰₯1 have |fn(x)|≀Mn
  2. βˆ‘n=1∞Mn<∞
then βˆ‘n=1∞fn converges uniformly.

Example 7.19.

The series βˆ‘n=1∞1n2+x2 converges uniformly over x∈R.
Solution.
Let Mn=1n2. Then |1n2+x2|≀1n2 for all x∈R and βˆ‘n=1∞1n2=Ο€26<∞, so by the Weierstass M-test, βˆ‘n=1∞1n2+x2 converges uniformly over x∈R.
For any r>0, the series βˆ‘n=0∞znn! converges uniformly on {z∈C:|z|≀r}, but βˆ‘n=0∞znn! does not converge uniformly on C.