Questions tagged [absolute-convergence]

Question 1

Is it possible to determine the convergence of a series $\sum a_n$ by evaluating $$\lim_{n \to \infty} \log_n(a_n)$$ and comparing the result to $-1$? Specifically, if this limit is less than $-1$, ...

Question 2

Here is the definition of an analytic map. ANALYTIC MAP Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if $$ f(x)=\sum_{k=0}^{\infty}a_k(x) \tag 1 $$ where for ...

Question 3

I was doing a problem where I was asked to show the continuity of $$\sum_{n = 1}^{\infty} e^{-nx}\sin{nx},$$ for $x>0$. My approach was to consider the sequence $(\sigma_{n}(x))$ of partial sums of ...

Question 4

Suppose that $\sum_{n = 0}^\infty a_n$ is absolutely convergent and $\{P_1,\ldots, P_r\}$ is a partition of $\mathbb{N}$ (i.e. $\bigcup_{i=1}^r P_i = \mathbb{N}$ and $P_i \cap P_j = \emptyset$ for any ...

Question 5

At a fairground, one big prize is being given away by a random chance game, such as spinning a wheel. A finite number of players line up in a queue and take turns with the game. If the person at the ...

Question 6

Consider this classic proposition for nets from Banach Algebra Techniques in Operator Theory by Douglas: 1.8 Definition Let $\{f_{\alpha}\}_{\alpha \in A}$ be a set of vectors in the Banach space $\...

Question 7

Examine the absolute/conditional convergence of the series $$ \sum_{n=1}^\infty \Bigl(\tfrac{\arctan n}{n} + (-1)^n\Bigr)\,\sin\!\Bigl(\tfrac1n\Bigr) $$ I can't seem to bound it from above, so I'm ...

Question 8

What is the best way to describe the relation between the ring of formal power series $\mathbb C[[X]]$ and the actual complex power series? I understand both separately, but I have never seen them ...

Question 9

I've recently been teaching myself about Cauchy sequences and I'm trying to understand a certain proof that the decimal expansion of $\sqrt{2}$ is a Cauchy sequence. This proof uses the following ...

Question 10

I am trying to prove a comment made on page 18 in Halmos's Introduction to Hilbert Spaces and The Theory of Spectral Multiplicity. For below, assume $X$ is an arbitrary Hilbert space. I am rephrasing ...

Question 11

I'm writing because I suspect Conway's "Functions of one complex variable" makes a minor mistake with its terminology, but I'm tired and unsure if I'm missing something. In the chapter on ...

Question 12

I am reading "Introduction to Analysis 1" (in Japanese) by Sin Hitotumatu. This book contains the following theorem. I found this theorem interesting. For example we can use this theorem ...

Question 13

The following proof of the rearrangement theorem is from Bartle and Sherbert's Introduction to Real Analysis, 3rd edition. I do not see where absolute convergence was used in the proof. 9.1.5 ...

Question 14

For $\mathop\sum\limits_{{n = 1}}^{\infty }{a}_{n}$, if for any subsequence $\left\{ {a}_{{n}_{k}}\right\}$, we have the convergences of $\mathop\sum\limits_{{k = 1}}^{\infty }{a}_{{n}_{k}}$ then by ...

Question 15

I came across the definition of absolute convergence of a series of functions and I am unsure if I did understand said definition correctly. Thus far, the only notion of absolute convergence I knew ...

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Questions tagged [absolute-convergence]

Is "the nth logarithm test" for series' convergence reliable?

Unconditional convergence and analytic map

Continuity of $\sum_{n = 1}^{\infty} e^{-nx}\sin{nx}$.

Split a series in a sum of series

Probability with a Converging Infinite Sum

Absolute convergence of a series in a Banach space.

Absolute/conditional convergence of $ \sum_{n=1}^\infty \Bigl(\tfrac{\arctan n}{n} + (-1)^n\Bigr)\,\sin\!\Bigl(\tfrac1n\Bigr) $

From formal power series to analytical ones

Showing the decimal expansion of $\sqrt 2$ is a Cauchy sequence

Two definitions of summable series in a Hilbert space

Converging absolutely and uniformly versus converging uniformly absolutely [duplicate]

Please suggest a book that explains this theorem. We can use this theorem when we prove $e^{iz}=\cos z+i \sin z$. (Sin Hitotumatu's analysis book.)

Why is absolute convergence necessary for the Rearrangement Theorem?

Absolute convergence of $\mathop\sum\limits_{{n = 1}}^{\infty }{a}_{n}$

Absolute convergence of a series of functions

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