1
vote
4answers
60 views

Why do we assign values to divergent series? [duplicate]

Why do we assign values to divergent series? For example, the series $1+2+3+4... = -1/12$. I understand the proof for this, but I feel like it uses false math, and I recall reading that you can't do ...
1
vote
1answer
58 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
2
votes
2answers
39 views

Very slow convergence of a particular series?

I've read that $$ \sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots $$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
3
votes
2answers
25 views

Another Divergent Series Question

Suppose the series $$\sum{a_n}$$ diverges where $a_n\ge 0$ and the sequence is monotone non-increasing. If exactly one element is chosen from each interval of size $k$ -- i.e., one element from ...
0
votes
1answer
24 views

Divergent Infinite Series Question

If the infinite series $$\sum{a_n}$$ diverges where all terms are positive and $\lim{a_n}=0$, is it always possible to construct a subsequence such that its series converges? That is, does there ...
3
votes
1answer
29 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
2
votes
4answers
61 views

How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $

I have put his on wolfram and obtained answer as follows: $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$ And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$ However I am ...
2
votes
0answers
32 views

Does this condition imply convergence of $\sum_k x_k$?

Let $\{x_n\}_{n \geq 1}$ be a sequence of real numbers (in principle, they could be complex numbers, but I don't think this makes much of a difference for this problem). Suppose the sequence $x_n$ ...
1
vote
3answers
39 views

Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$

For $a_n \geq 0$, and $\sum a_n$ convergent, show that $\sum \frac{\sqrt{a_n}}{n^p}$ is also convergent for $p > 1/2$? What bugs me more is why isn't $\sum \sqrt{\frac{a_n}{n}}$ convergent?? ...
2
votes
1answer
23 views

Convergence almost everywhere

Let consider rational numbers $\{r_n\}_{n=1}^{\infty}$ on [0, 1]. How to prove, that such sum $$\sum_{n=1}^{\infty}\frac{1}{n^2|x-r_n|^{0.5}}$$ converges almost everywhere on [0, 1]. There are my ...
0
votes
2answers
27 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
1
vote
1answer
33 views

Prove that the following sequence converges and find its limit

Prove that the following sequence converges and find its limit $$\frac{1}{3+\frac{1}{3}},\frac{1}{3+\frac{1}{3+\frac{1}{3}}},\frac{1}{3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}},\ldots$$ I started to ...
0
votes
1answer
50 views

Suppose that $\sum a_i$ converges and that $a_i\geq0$ for all 𝑖.

Suppose that $\sum a_n$ converges and that $a_n\geq0$ for all $n$. For each $n$, let $e_n=\pm1$. Then, prove that $\sum e_na_n$ converges. Can I simply say that ∑|eᵢaᵢ| = ∑aᵢ so that ∑eᵢaᵢ converges ...
2
votes
1answer
48 views

Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the ...
0
votes
0answers
17 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
3
votes
3answers
117 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
0
votes
3answers
29 views

Convergence of an Alternating Series

For this question, I`m trying to determine the values of $p$ in which the series converges. The series is: $\sum_{n=1}^\infty \frac{(-1)^n}{n^p}$. I already know (by looking at the answer to the ...
1
vote
0answers
21 views

Series convergent but not absolutely? $\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$

For which real numbers $p>0$ does the series $$\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$$ converge? Obviously it converges absolutely for $p>1$ but what about $0<p<1$? I have the ...
-1
votes
0answers
22 views

An example of a sequence of continuous real functions pointwise convergent, but nowhere locally uniformly convergent? [duplicate]

I've been trying to come up with an example of a sequence of continuous real function which would converge pointwise everywhere, but nowhere converge locally uniformly, but I can't really think of ...
2
votes
2answers
53 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
1
vote
3answers
76 views

Simple series convergence/divergence

I have the following problem: $$\sum_{k=1}^{\infty}\frac{2^{k}k!}{k^{k}}$$ I only need to find whether the series converges or diverges. My initial thinking was to use the ratio test. I hit a stump ...
0
votes
1answer
35 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
1
vote
2answers
39 views

Testing A Series For Convergence

Determine whether the series $\sum_{n=0}^{\infty} \frac{3n^2 + 2n + 1}{n^3 + 1}$ with n from 0 to infinity converges or diverges. So far I thought about dividing the numerator by the denominator, ...
0
votes
4answers
129 views

Why do some series converge and others diverge? [closed]

Why do some series converge and others diverge? What causes the divergence or convergence of a series and why does that cause such a behavior? For example, why does the harmonic series diverge, but ...
1
vote
3answers
43 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
2
votes
2answers
77 views

Convergence of $ \sum_{n=1}^{\infty} (\frac{n^2+1}{n^2+n+1})^{n^2}$

Find if the following series converge: $$\displaystyle \sum_{n=1}^{\infty} \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ What I did: $$a_n=\left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ $$b_n=\frac ...
1
vote
1answer
28 views

Let $ \sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ \sum_{n=1}^{\infty}2(a_n)^3$ converges as well

Let $ S_1=\displaystyle\sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ S_2=\displaystyle\sum_{n=1}^{\infty}2(a_n)^3$ converges as well. We have $\exists l :\forall ...
0
votes
0answers
18 views

Consider the sequence $\{p_n\}_{n \in P}$… [closed]

Consider the sequence $S=\{p_n\}_{n \in P}$, where $p_n$ is the decimal expansion of $\sqrt{2}$ truncated at the $n$th decimal place (so $p_1=1.4, p_2=1.41, p_3=1.414$, etc.) If we work only in ...
1
vote
1answer
28 views

Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
1
vote
0answers
35 views

Gauss' test for Convergence

In the text I am using, the hypotheses used for Gauss' test for convergence are different to others I have seen. The text has the hypotheses: if the series $\sum_{n=1}^{\infty} a_n$ is such that ...
4
votes
1answer
56 views

Different types of Convergence for a Series Function

I am currently investigating the convergence of the following function, $f(x)=\sum\limits_{k=1}^{\infty} \dfrac{x^{k}+\sin(k)}{k^{2}}$ for different "senses". I have shown that $f(x)$ converges ...
1
vote
1answer
39 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
2
votes
2answers
32 views

Two series sum, one converging, one diverging

I need to give an example of a two series sum a_n and b_n such that the lim a_n/b_n=1. One series has to diverge and one has to converge. a_n and/or b_n don't necessarily have to be positive. I have ...
1
vote
2answers
33 views

Connection between series

I have to show that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} a_n^2$ is absolutely convergent too. Please give me some hint, how do I start the excercise. ...
0
votes
1answer
24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...
1
vote
1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
2
votes
2answers
84 views

If $\sum a_n$ converges then $\sum (-1)^n \frac {a_n}{1+a_n^2}$ converges?

Could you please give me some hint how to deal with this question: If $\sum a_n$ converges, does this necessarily mean that $\sum (-1)^n \frac {a_n}{1+a_n^2}$ must converge also ? Thanks.
4
votes
2answers
34 views

Help with Convergence/Divergence

So I am trying to prove whether the following problem converges or diverges? $$\sum_{n=1}^\infty \left({n\over n+18}\right)^n$$ So I decided to use the Root test. $$ L = \lim_{n\to ...
0
votes
0answers
28 views

Prove that $g_n \to g$ uniformly on $\mathbb{R}$ but $f_{n}g_n$ foes not converge uniformly to$fg$ on $\mathbb{R}$

Let $f_n(x) = x$ and $f(x) = x$ for $x \in \mathbb{R}$ and let $g_n(x) = \frac{1}{n}$ and $g(x) = 0$ for $x \in \mathbb{R}$. Prove that $f_n \to f$ uniformly on $\mathbb{R}$ and $g_n \to g$ uniformly ...
0
votes
0answers
26 views

Continuity of convergence vector

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i := ...
1
vote
0answers
42 views

Change of signs in harmonic series

Prove that if we choose signs for individual terms in harmonic series $\sum_{n=1}^{\infty}{1\over n}$ in such a way that $p$ positive terms are followed by $q$ negative terms (without rearranging the ...
1
vote
2answers
40 views

Convergence of $\sum_{k=1}^n(1-k/n)a_k$

Assume that the series $\displaystyle \sum_{n=1}^\infty a_n$ converges to a finite number, say $S$. Now let's consider a sequence of modified partial sums $\displaystyle ...
0
votes
1answer
22 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
2
votes
2answers
59 views

A convergent series property [duplicate]

This came up in a friend's exam and it must be one of those ${\epsilon},N(\epsilon)$ arguments I could do in a snapshot in my twenties but now I can't figure out how the proof should go: For a ...
1
vote
0answers
18 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
2
votes
1answer
41 views

Infinite Series: Convergent?

I just came across the following question from an old book: $$S=\sum_{k=1}^{\infty} \frac{\cos(\sqrt{k} \pi)}{\sqrt{k}}.$$ Is it convergent? My guess is that it is but how to show it? Thanks in ...
2
votes
1answer
35 views

Convergence of series $A_n = \sqrt{\sum_{k=n}^{\infty} a_k} - \sqrt{\sum_{k=n+1}^{\infty}a_k} $ if series $a_n$ converges

I must show that if a series $\sum_{n=1}^{\infty} a_n$ with positive terms converges, then the series $\sum_{n=1}^{\infty} A_n$, where $A_n = \sqrt{\sum_{k=n}^{\infty} a_k} - \ ...
0
votes
1answer
18 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
0
votes
3answers
48 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
1
vote
2answers
135 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...