MHB Testing Uniform Convergence of Complex Function Sequences with Natural Numbers

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To determine uniform convergence of the given sequences of complex functions, one must first identify the pointwise limits for each sequence. For the sequence z/n, the limit is 0 as n approaches infinity for any fixed z. The sequence 1/nz converges to 0 for z ≠ 0, while for nz^2/(z+3in), the limit depends on the behavior of z as n increases. The analysis should specify the set over which uniform convergence is being tested, as this impacts the results. Understanding the domains of convergence is crucial for accurately assessing uniform convergence in these cases.
Poirot1
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How do I determine whether the following sequences of complex functions converge uniformly?

i) z/n

ii)1/nz

iii)nz^2/(z+3in)

where n is natural number
 
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You have to specify on which set you want uniform converge (except if you have to determine the domains of convergence). First, find the pointwise limits.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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