7.23
7.23 Where do the following sequences converge pointwise? Do they converge uniformly on this domain?
(a) \((nz^n)\)
(b) \((\frac{z^n}{n})\)
(c) \((\frac{1}{1+nz})\) where Re\((z) \geq 0\)
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(a) \((nz^n)\) converges to 0 where \(\mid z \mid < 1\), diverges to \(\infty\) where \(\mid z \mid = 1\), and diverges to \(\infty\) where \(\mid z \mid > 1\).
The sequence converges uniformly on the interval \((0,1)\)

(b) \((\frac{z^n}{n})\) diverges for all values of \(\mid z \mid\).
Cannot converge uniformly since the sequence does not converge.

(c) \((\frac{1}{1+nz})\) where Re\((z) \geq 0\) converges to 1 (constant function of \(\frac{1}{1}\) when Re\((z) = 0\) and diverges for all values when Re\((z) > 0\).
The sequence converges uniformly only when Re\((z) = 0\).