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Taylor and Laurent Series — Howell & Matthews
Mathematics — Complex Analysis · uniform-convergence
Problem Prove that the following series converge uniformly on the sets indicated.
((a)) ∑ k = 1 ∞ 1 k 2 z k \displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}z^k k = 1 ∑ ∞ k 2 1 z k D ‾ 1 ( 0 ) = { z : ∣ z ∣ ≤ 1 } \overline{D}_1(0) = \{z : |z| \leq 1\} D 1 ( 0 ) = { z : ∣ z ∣ ≤ 1 } ∑ k = 0 ∞ 1 ( z 2 − 1 ) k \displaystyle\sum_{k=0}^{\infty}\frac{1}{(z^2-1)^k} k = 0 ∑ ∞ ( z 2 − 1 ) k 1 { z : ∣ z ∣ ≥ 2 } \{z : |z| \geq 2\} { z : ∣ z ∣ ≥ 2 } ∑ k = 0 ∞ z k z 2 k + 1 \displaystyle\sum_{k=0}^{\infty}\frac{z^k}{z^{2k+1}} k = 0 ∑ ∞ z 2 k + 1 z k D ‾ r ( 0 ) \overline{D}_r(0) D r ( 0 ) 0 < r < 1 0 < r < 1 0 < r < 1 Browse the problem library