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Please look at how the verb disturb is used in this sentence:

This example suggests that for a power series, the uniformity of convergence may at most be disturbed near the circle of convergence.

This is the example mentioned. Because it contains mathematical expressions and I cannot type that here, I will post its picture enter image description here

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I referred to the dictionary but could not find the right meaning. I only know that the convergence is uniform within the convergence circle but uniform convergence does not hold on the circle itself.

The second part of my question is related to the use of the word setting in this sentence:

We now shift from power series to arbitrary series of variable terms and examine uniform convergence in this more general setting.

I know that the author's intention of ''this more general setting'' is that we consider an arbitrary series, not just a power series. Now, what is the appropriate and precise meaning of ''setting'' in this sentence?

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    You have two distinct questions here, both of which can be looked up in a dictionary. In my dictionary, the correct interpretation is interpretation 1 in both cases. Apr 25, 2023 at 18:15
  • It is not polite, and may be counterproductive, to order us to 'pay attention' to something. Apr 25, 2023 at 19:35
  • @DoneWithThis. I have seen those meanings in different dictionaries, but I feel that they are not exactly suitable for this text Apr 25, 2023 at 20:37
  • @MichaelHarvey I am sorry. I use Google to translate my questions. Some words may not be translated correctly and may sound impolite. Apr 25, 2023 at 20:39
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    @absolutezero The preferred way on ELL is for you to find the definition yourself, include it in the question, and explain why it didn't resolve the problem for you. You'll likely get better answers, among other benefits. Apr 25, 2023 at 23:42

1 Answer 1

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  1. ...the uniformity of convergence may at most be disturbed near the circle of convergence.

The appropriate dictionary definitions are (from Merriam-Webster)

1.a to interfere with: interrupt
2.b to throw into disorder

The sense of the statement is that, near the circle of convergence, the series retains the property of uniform convergence but not in a natural way. Indeed, near the circle of convergence, the series does converges uniformly, but only in a very particular, somewhat unnatural sense: it converges uniformly in any closed disc contained within the (open) circle of convergence. The author has chosen the word disturb because it suggests that the situation is more complicated but the property is preserved. (In my opinion the author's phrasing is somewhat odd).


  1. We now shift from power series to arbitrary series of variable terms and examine uniform convergence in this more general setting.

The relevant dictionary definition is (again from Merriam-Webster)

3.a the time, place, and circumstances in which something occurs or develops

In this case circumstance is the key word. The author has been developing a theory of convergence in the context of power series; now the author is going to continue developing the theory by taking into account more general series (i.e. without an x^n term). Since power series are a special type of series, this new setting is more general.


PS. I just wrote an analysis exam this morning, lol

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