For questions about recurrence relations, convergence tests, and identifying sequences

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1
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4answers
47 views

Maclaurin series of function $f(x)= x+\ln(9-x^2)^\frac{1}{3}$

Find the Macluarin series. I'm trying for hours to understand how should I solve it. Please explain it to me step by step. $f(x)=x+\ln {\sqrt[3]{9-x^2}}$
0
votes
2answers
26 views

How do I prove this statement about limits?

If $a_n$ is monotonic increasing and $b_n$ is a Bounded series and the limit of $a_n$ - $b_n$ is zero then prove that $b_n$ has a limit.I know that if I proved that $b_n$ is monotonic ...
0
votes
2answers
23 views

Norm on the space of sequences

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, I'm trying to show that ...
1
vote
2answers
46 views

$\sum_{n=2}^\infty {(-2)^n \over n} $ How does this converge or diverge using the alternate series test?

$$\sum_{n=2}^\infty {(-2)^n \over n} $$ When I took the limit I got -2, I also tried using ratio and root test and got the same answer. The answer is supposed to be divergent I think but I thought if ...
4
votes
1answer
50 views

If $\limsup_n\sqrt[n]{a_n}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{(n+1)a_{n+1}}=?$

If $\limsup_n\sqrt[n]{|a_n|}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{|(n+1)a_{n+1}|}=?$ Can I separate the product of the second limit as ...
0
votes
1answer
54 views

Sum of hyperbolic functions, having problems expressing $\sinh(1)$

Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = (-1+\cosh(1))\cosh(x) + \sinh(1)\sinh(x)$ Express the series $C = \cosh 0 + ...
5
votes
7answers
104 views

Show convergence/divergence for $\sum_{n=1}^{\infty} \frac{{(\ln n)}^{2}}{{n}^{2}} $

$$\sum_{n=1}^{\infty} \frac{{(\ln n)}^{2}}{{n}^{2}} $$ Anyone can give hint for this? Thank you!
1
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1answer
51 views

Limit comparison test

Since the series $\sum_{n=1}^\infty a_n$ is convergent, so $\lim_{n\to\infty} a_n=0.$ Consider $$\lim_{n\to\infty} {\left(\dfrac{{a_n}^{1/2}}{n}\right)\over{\left(\dfrac{1}{n^2}\right)}} $$ which is ...
2
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3answers
44 views

Finding the Limit of the Ratio of a Recursive Sequence's Terms

Let {$f_n$} be defined recursively as $f_1 = f_2 = f_3 = 1$ and $f_n = f_{n-1} + f_{n-3}$ for all $n \gt 3$. Also, define {$a_n$} as the ratio of the terms of {$f_n$}. That is, $a_n = ...
2
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2answers
46 views

How to prove convergence of the Arithmetic-Geometric Mean sequence?

Let $$a_0>b_0>0 $$and consider the infinite sequences $$\{a_n\}, \{b_n\}$$ where $$a_{n+1}=\frac{a_n+b_n}{2}$$ and $${b_{n+1}}={(a_nb_n)}^{1/2} $$ for $n\geq0$. Prove that the infinite ...
0
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0answers
19 views

Requesting help on understanding series [on hold]

Is the tangent of a positive convergent series still positive?
-2
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0answers
50 views

a friend posted this on facebook i need help to figure it out [on hold]

She posted this fraction on facebook five days ago 148/13. And then yesterday she posted 145/15 and today she posted 145/18 she said the goal is a whole number what would that number be?
0
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0answers
26 views

Number of Unique Permutations of 3 digits (-1,0,1) given a length that match a sum

Say you have a vertical game board of n length (length being number of spaces). And you have a three sided die that has the options: go forward one, go back one, and stay. If you go below or above ...
-1
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1answer
32 views

I have a feeling that these statements on my homework are true, but how would I prove it? [on hold]

If $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are both absolutely convergent series with all positive terms then $\sum_{n=1}^{\infty}a_n/b_n$ is absolutely convergent. If the power ...
1
vote
1answer
43 views

A little help proving this statement?

If $a_n$ is a sequence that implies $a_{n+1}\lt a_n$ and $b_n$>0 is a sequence which implies $b_{n+1}\gt b_n$ and also $b_{n+1}=\sqrt{a_n\cdot b_n}$ then I need to prove that $a_n$ and $b_n$ has a ...
2
votes
2answers
40 views

How should I go about finding convergence/divergence of these two series?

$$ \sum_{n=0}^\infty {2^n \over 3^n +1} $$ $$ \sum_{n=0}^\infty {n^4 \over 4^n} $$ The first one I think could be a geometric series, but the +1 in the denominator is confusing me. For both I'm not ...
1
vote
1answer
26 views

How to prove this statement about Sub-sequences?

How to prove that if the sub-series of the even numbers (the elements which has an even index) has a limit and the sub-series of the odd numbers (the elements which has an odd index) have the same ...
1
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0answers
25 views

Calculate infinite sum

I would like to find a closed for the following sum: $\sum_{i=1}^\infty a^{i^\beta}$ Where $|a|<1$, and $\beta\geq 1$. Anyone any ideas? Best, Ben
0
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3answers
57 views

How to find if $\lim_{n\to \infty} {n \over 3^n}$ is convergent or divergent

$$\lim_{n\to \infty} {n \over 3^n}$$ I think this involves the comparison test? My thought was comparing it to $1 \over 3^n$
0
votes
5answers
57 views

How can I find if the sequence $z_n = \sqrt{n+2} - \sqrt{n} $ converges or diverges?

$$z_n = \sqrt{n+2} - \sqrt{n} $$ $$ \lim_{n\to \infty} z_n =\lim_{n\to \infty} \sqrt{n+2} - \sqrt{n} $$
0
votes
0answers
21 views

Given $u_{n+1}=u_n+\frac{1}{n^\alpha u_n}$. Show that $u_n\leqslant \sum_{k=1}^{n}\frac{1}{k^\alpha u_1}$?

Let $u_{n+1}=u_n+\dfrac{1}{n^\alpha u_n}$ for all $n\geqslant1$ and $u_1>0$. Show that for all $n\geqslant 1$: $$\displaystyle u_n\leqslant \sum_{k=1}^{n}\dfrac{1}{k^\alpha u_1}.$$ I found that ...
0
votes
2answers
32 views

Calculating a power series [on hold]

I was wondering if anyone knows how to calculate: $\sum_\limits{t=-\infty}^{\infty}$ $a^{t} e^{-itb}$, for constants a,b and $-\pi < b < \pi$ Can we take the t=0 term out to reduce it ...
2
votes
2answers
89 views

the lim of sum of sequence

I have to calculate the following: $\lim_{n \to \infty} (\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} )$ I managed to understand that it is $\lim_{n \to \infty} ...
1
vote
3answers
59 views

Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$

Do you know how to find the limit of $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$ and m is a natural number? I've tried to google it in wiki but I do not understand the closed form ...
0
votes
0answers
43 views

Infinity Series [on hold]

Good afternoon. I'm brazilian, then sorry by my bad english. I have a problem with one question about Infinite Series. I searched for anyone method could help me. I have all constants values (w, y, ...
1
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0answers
31 views
+50

An upper bound for $\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$.

Let us consider $a_n, \phi_n, K_{n}$ complex sequences. Let $$\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$$ where $\left|K_{n}\right|\leq \gamma$ and $\gamma>0$. Can we ...
1
vote
2answers
64 views

Is my proof of a seemingly trivial question involving infinite sums correct?

Given that $\sum_{n=0}^{\infty}a_n^2<\infty$ and $\sum_{n=0}^{\infty}b_n^2<\infty$, where $a_n$ and $b_n$ are sequences of real numbers, I'm trying to show that ...
2
votes
2answers
41 views

Serie involving Bernoulli's numbers

I need to find the exact sum of this serie which involves Bernoulli's numbers: $$\sum_{k=1}^\infty {{B_{2k}(k-1)!\over (2k)!}}$$ It converges very quickly but I'm knew to this kind of problems so I ...
4
votes
1answer
46 views

Sums of reciprocals of subsets of natural numbers

There exists such a subset $A$ of the reciprocals of natural numbers $\{\frac{1}{n} \ |\ n \in \mathbb N\}$ that any real number $x$ on the interval $[0,1]$ can be expressed as sum of members of some ...
1
vote
1answer
11 views

Describing convergence/divergence of a complex sequence

Let (a$_n$)$_{n \in N}$ be a complex sequence and a $\in$ C. Show that the following statements are equivalent: $\forall$ $\varepsilon$ > 0 $\exists$ N $\in$ N $\forall$ n $\geq$ N : |a$_n$ - a| ...
1
vote
1answer
21 views

Name of a particular condition on sequences

I've come across a certain condition on a sequence that I figure I'm probably not the first to come across, but I can't work out what it's called. If the sequence has two elements $(a,b)$, then the ...
1
vote
2answers
28 views

What is the example of a not almost convergent sequence but whose Cesàro means converge?

It seems to me a sequence that is almost convergent implies that its Cesàro means converges but not vice versa. What is the example that a not almost convergent sequence whose Cesàro means converge. ...
0
votes
1answer
16 views

sequence spaces as subsets of each other

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, how can it be shown ...
0
votes
1answer
43 views

Limit of a convergent serie

For a research project, after some manipulation I come up with a convergent serie that I have to prove its limit. The statement is the following: $ \lim_{n \rightarrow \infty } \displaystyle ...
2
votes
1answer
32 views

Sequence in product metric space [on hold]

Let we have $(X_1,d_1)$ is a metric space and $(X_2,d_2)$ is another metric space . Now we will difend $X=X_1*X_2$ and we have $d$ is a distance function on $X$ So $(X,d)$ is a metric Space I ...
0
votes
2answers
41 views

Proof: Convergence of a series implies convergence of another series

Question: $$\sum_{n=1}^\infty a_n $$ is convergent with $a_n$ positive,prove the series $$\sum_{n=1}^\infty \frac {{(a_n)}^{1/2}}{n}$$ is convergent. The hint given is: $x^2+y^2\geq2xy$. But I ...
0
votes
1answer
14 views

How to show $[0, \frac{\pi}{2}) \cup (\frac{3\pi}{4}, 2\pi)$ is sequentially separated?

Definition of separated: $E$ is separated if $A, B \neq \varnothing$, $A \cup B = E$, and there are not convergent sequences in $A$ that have limit points in $B$, and vice versa. Definition of ...
6
votes
3answers
107 views

Find the sum of the n first series numbers: $7,77, 777,…$

Find the sum of the $n$ first numbers: $7,77, 777,...$ I thought to find an order by dividing $77/7=11, 777/7=111...$ but I don't know how to continue.
4
votes
3answers
104 views

Which (convergent) series can one find the sum of?

I know about geometric series and how one can find the sum when they are convergent. I also have heard that one can prove that the $p$-series $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ has sum $\pi^2/6$ ...
1
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0answers
62 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
1
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0answers
31 views

Two statements about partial limits and intervals

Let $(a_n)$ be a sequence and $I=(a,b)$ an open interval such that every $L\in(a,b)$ is a partial limit of $(a_n)$. Decide whether or not the following statements are true: $\{a_n | n \in ...
1
vote
5answers
66 views

How can I compute $\sum_{n=0}^{\infty} 0.6^n$? [duplicate]

I am a computing teacher and just helping out some students with a math question. They have been asked to calculate the following: $$\sum_{n=0}^{\infty} 0.6^n$$ I am intrigued as to how one gets to ...
5
votes
9answers
111 views

If $a_n = \frac{e^{n}}{e^{2n}-1}$ how do I show that $a_{n+1} \leq a_n$?

Let $$a_n = \frac{e^{n}}{e^{2n}-1}$$ How do I show that $a_{n+1} \leq a_n$? I don't know how to deal with the $-1$ in the denominator.
0
votes
0answers
34 views

Which sum formula should I use for arithmetic sequences?

I was looking online for a formula that would allow me to find the sum of arithmetic sequences and I came across the 2 that I have listed below. I tried to use both them and they worked. So, I ...
0
votes
1answer
30 views

Recursive Sequence with different conditions

I'd like to deduce a formula for a slightly wierd recursive sequence i've got $$ f(n) = \begin{cases} f(n - 1 ) + 1, & \text{if $n$ is even} \\ 2f(n - 1), & \text{if $n$ is odd} \end{cases} ...
1
vote
1answer
41 views

If a sequence is decreasing to zero, why do we have the following

Let $\{a_i\}_{i\in\mathbb{Z}}$ be some real sequence and let $S_n = \frac{1}{n}\sum_{|i|<n}^n|a_i|$. I need to verify two claims. First, if $a_n\to0$ as $n\to\infty$, why ...
1
vote
3answers
42 views

Question Sequences and Series

Prove that the sequence $(a_n)_{n \geq 1}$ defined by $$ a_n= \sum_{k=1}^{n} \frac{1}{k} - \ln(n+1)$$ is increasing and bounded. It is on the study guide for my final exam, which is tomorrow so I am ...
1
vote
2answers
37 views

Limits and sequences question

Let $$a_n=n^x(n^{1/n^2}−1).$$ Show that $$\lim_{n \to \infty} \frac{a_n}{\ln(n)/(n^{2-x})} = 1. $$ It is on the study guide for my final exam, which is tomorrow so I am trying to figure it out. ...
0
votes
0answers
21 views

Series and sequences- Help [on hold]

Let $a_n=n^x(n^(1/n^2)-1)$ for n in natural numbers and assume that lim(n goes to infinity)ln(n)/n^r = 0 for any r>0. Let ln(x)=integral(from t=1 to x)dt/t for x>0. Prove the inequality h/1+h < ...
3
votes
3answers
57 views

Series Proof $\sum_{k=1}^n (1/k) > \ln(n+1)$

Prove that $\sum_{k=1}^n (1/k) > \ln(n+1)$. I have been trying to do this for some time now, but I cannot figure it out. It is on the study guide for my final exam, which is tomorrow so I am trying ...