Absolute and uniform convergence of <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXA

hushjelpw4

hushjelpw4

Answered question

2022-05-27

Absolute and uniform convergence of n = 1 n = 2 n sin 1 3 n z

Answer & Explanation

Alberto Duffy

Alberto Duffy

Beginner2022-05-28Added 5 answers

To get the absolute convergence, just use the inequality | sin t | | t | for any real number t, which be established thanks to the fundamental theorem of analysis:
| sin t | = | 0 t cos s d s | 0 | t | cos s d s | t |
Let S a subset of C of the form S := { z = 0 < | z | < δ }, we have to show that the convergence is not uniform on S. To see that, note that for n large enough, say larger than n 0 contains points of the form 3 n , so sup z S | n = N + 2 n sin 1 3 n z | 2 N sin 1 , since the terms for z = 3 n are non-negative if n n 0

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