Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
63,892
questions
-2
votes
0
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11
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Find the value of question mark form the given information.
Find the value of question mark form the given
-1
votes
1
answer
26
views
Space where normal convergence does not imply uniform convergence [closed]
We know that if we're in a Banach space, normal convergence implies uniform convergence. Is there an easy counterexample outside Banach spaces ?
- 88
2
votes
0
answers
39
views
Series with more terms converges more slowly than with lesser terms
I was working with the Laplace differential equation with certain boundary conditions, problem that yields the following analytic solution:
$$
\mathbb{V}(x,y)= \mathbb{V}_o\frac{4}{\pi}\sum^\infty_{n=...
- 147
0
votes
0
answers
31
views
Calculating the value of $\mathbb ln(3.01+\sqrt{4.12}) $
My goal is to calculate $$\mathbb z= ln(3.01+\sqrt{4.12}) $$ with the prior knowledge that the data was obtained via truncation.
But how can I use $\mathrm{|R_n|=|S-Sn|}$, where $\mathrm{S_n}$ is the ...
- 35
1
vote
0
answers
28
views
Variation of a sequence of approximations : $u_n = \lfloor nx\rfloor/n$
Given a real number $x$, define the sequence $u$ by :
$$\forall n\in\mathbb{N},\quad u_n=\frac{\lfloor 10^nx\rfloor}{10^n}$$
It is well known that $\lim_{n\to\infty}u_n=x$ and that sequence $u$ is non-...
- 7,283
1
vote
0
answers
88
views
How do we calculate this infinite summation
How do we evaluate this summation:
$$H(b, c)=\sum_{n=1}^\infty \frac {\cos(c \log(n))} {n^b}$$ where $b, c$ are some positive constants such that $0<b<1$.
I know (if I am correct) that this ...
0
votes
2
answers
45
views
$\lim\limits_{n \to \infty} \lim\limits_{m \to \infty} a_{mn}=\lim\limits_{m \to \infty} \lim\limits_{n \to \infty} a_{mn}=0$ implies $a_{nn} \to 0$?
The question is as in the title.
Let us suppose that a double sequence $(a_{mn})$ satisfies
\begin{equation}
\lim\limits_{n \to \infty} \lim\limits_{m \to \infty} a_{mn}=\lim\limits_{m \to \infty} \...
- 6,820
0
votes
1
answer
32
views
Question about convergence and divergence of positive series,which means there exists no slowest convergent series.
If positive series $\sum_{n=1}^{\infty} a_n$ is divergent,try to prove exist divergence
positive series $\sum_{n=1}^{\infty} b_n$,which satisfies
$\lim _{n \rightarrow \infty} \frac{b_n}{a_n}=0$
If ...
1
vote
0
answers
166
views
Prove the following equality: $1=\dfrac{4}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{7}\cdot\dfrac{10}{9}\cdot\dfrac{12}{11}\cdots$
The product can be transformed into:
$1=\left(1+\dfrac{1}{3}\right)\cdot\left(1-\dfrac{1}{5}\right)\cdot\left(1-\dfrac{1}{7}\right)\cdot\left(1+\dfrac{1}{9}\right)\cdots$
The result after a few ...
- 491
4
votes
0
answers
41
views
Ramanujan's identity concerning a quotient of Dedekind's eta functions
In his paper On certain Arithmetical Functions (published in Transactions of the Cambridge Philosophical Society, XXII, No. 9, 1916, pp. 159-184) Ramanujan presents the following identities (as if ...
- 85.2k
0
votes
1
answer
58
views
Proving that if $s_n=\frac1n\sum_{k=1}^n a_n$ diverges to infinity, then $a_n$ also diverges to infinity.
I am struggling to prove that a sequence $(a_n)$ diverges to infinity if its sequence of averages $s_n=\frac1n\sum_{k=1}^n a_n$ diverges to infinity.
My attempt:
We prove the contrapositive. Let $N\in\...
- 219
4
votes
0
answers
81
views
A Series of Integrals like $\sin (x^2)/x^2$
Is there a general solution or pattern in the following series of integrals?
$$S(n)=\idotsint_{-\infty}^{\infty} \frac{\sin(x_1^2+x_2^2+...+x_n^2)}{x_1^2+x_2^2+...+x_n^2}\, dx_1 \dots dx_n$$
I can ...
- 375
0
votes
1
answer
36
views
Are there any sequences whose sequence of averages $s_n=\frac1n\sum_{k=1}^n a_n$ diverges to infinity but $a_n$ is bounded?
I am struggling to find any examples of sequences $a_n$ whose sequence of averages $s_n=\frac1n\sum_{k=1}^n a_n$ diverge to infinity but $a_n$ is bounded. Perhaps, there is no such sequence?
- 219
0
votes
1
answer
59
views
For what $x\in\mathbb{R}\setminus\{0\}$ does generalized continued fraction $x+\frac{x}{x+\frac{x}{x+\frac{x}{x+...}}}$ converge?
For what $x\in\mathbb{R}\setminus\{0\}$ does generalized continued fraction $x+\frac{x}{x+\frac{x}{x+\frac{x}{x+...}}}$ converge? Being an undergraduate student knowing only information from Wikipedia ...
- 162
0
votes
0
answers
26
views
Confusion about the definition of a nonlinear recurrence relation and recurrence relation with fractional subscript.
I've been taking a discrete mathematics course, and we've defined the following type of recurrence relation as nonlinear: $$T_{n} = \sum_{i=1}^k a_{i}T_{n/b_{i}}$$
where $k$, $a_{i}$ and $b_{i}$ are ...
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3
votes
1
answer
68
views
Does the sequence $x_n := n^2/\sqrt{n^6+1} + n^2/\sqrt{n^6+2} + . . . + n^2/\sqrt{n^6+n}$ converge? If it does, what value does it converge to?
Does the sequence converge? If it does, to what value?
In my solution,
I have rewritten the expression as $x_n = \frac{1}{\sqrt{n^2+\frac{1}{n^4}}} + \frac{1}{\sqrt{n^2+\frac{2}{n^4}}} + . . . + \frac{...
- 33
1
vote
1
answer
20
views
Equation of a sequence of a pyramid of glasses. Trying to find the sequence equations for the fill rates and number of pours.
This is my problem:
I am trying to find equations for the sequences in order to calculate the fill rate of a glass and how many pours until it starts to be filled.
For example (tower of the fill ...
- 13
0
votes
0
answers
48
views
why can I not use an alternating series test to find out if the sequence below converges or not?
$$\sum_{i=1}^∞ \frac{(-1)^i(i^3+1)}{i^3+7}$$ why can I not use the alternating series test on this series to determine convergence?
0
votes
0
answers
43
views
If subsequences of sequences of real numbers are uncountable then why are they represented as $\{x_{nk}\}$?
I've noticed that subsequences of a sequence $\{x_n\}$ in real numbers are represented as $\{x_{nk}\}$. Why is this done when the subsequences of sequences of real numbers are uncountable?
- 33
1
vote
1
answer
50
views
Problem in understanding proof of Alternating Series Theorem
I am having problems understanding the second part of the theorem. However, I will state the full theorem, as well as the full proof, since proof of the first part is used for the second part. I have ...
- 187
2
votes
0
answers
56
views
Derivation or proof of the Dirichlet L-series of order $1$
I would like to know how the Dirichlet L-series(of order $1$) were derived and/or proved. I independently found sequences analogous to the Dirichlet L-series using a property from: Interesting ...
- 491
0
votes
1
answer
49
views
Is it possible to separate a convergent series term-wise into two sub-series?
What are the rules for term wise separation of two(or the more general case of $n$) infinite series:
Given an infinite series $H=\sum_{n=1}^\infty f(n, a, b) - g(n, a, b)=C_0$ where $a, b, C_0$ being ...
2
votes
2
answers
81
views
Show $P_n = \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \dots + \frac{(-1)^n}{n!}$ given $P_n = \frac{n-1}{n} P_{n-1} + \frac1n P_{n-2}$
Show that $P_n = \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \dots + \frac{(-1)^n}{n!}$ given $P_n = \frac{n-1}{n} P_{n-1} + \frac1n P_{n-2}$, $P_1 = 0$, $P_2 = \frac12$
I have absolutely no clue ...
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1
vote
2
answers
127
views
Given that: $1=(1+\frac{1}{3})(1-\frac{1}{5})(1-\frac{1}{7})(1+\frac{1}{9})(1+\frac{1}{11})\cdots$, Prove the following:
Prove that $$(\dfrac{1}{1}+\dfrac{1}{3}-\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{9}+\cdots)\cdot(\dfrac{1}{1}+\dfrac{1}{3}-\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{9}+\cdots)=\dfrac{1}{1^2}+\dfrac{1}{3^2}+\...
- 491
-1
votes
0
answers
46
views
Decide if the following series converge and explain your reasoning: $\sum_{n=1}^{∞}\frac{\cos(2n)}{n^3}, \sum_{n=1}^{∞}\frac{1}{4^n + 2n}$ [closed]
(a) $$\sum_{n=1}^{∞} \frac{\cos(2n)}{n^3}$$
(b) $$\sum_{n=1}^{∞} \frac{1}{4^n + 2n}$$
I'm a little confused on how to approach this problem -- any tips? Thank you!
- 1
1
vote
1
answer
85
views
T\F: if $\lim_{k\to\infty}\sum_{i=n_k}^{n_{k+1}-1}|\alpha_i|= 0$ then $\sum \alpha_k$ converges.
Let $\sum \alpha_k$ be a bounded series of reals.
Suppose $\{n_k\}_{k=1}^{\infty}$ is a strictly-increasing monotonic sequence of naturals s.t $lim_{k\to\infty} (n_{k+1}-n_{k})= \infty$.
Prove or ...
- 877
0
votes
0
answers
23
views
Optimizing a recurrence relation for a sequence
Given the sequence $$a_k=\frac{(2k)!}{4^k(k!)^2(2k+1)}(0.5)^{2k+1},$$ I should find a recurrence relation for it. I came up with $a_0 = 0.5$ and $$a_{k+1}=\frac{(2k+1)^2}{8(k+1)(2k+3)}a_k.$$ is there ...
- 35
1
vote
0
answers
21
views
Equivalence of expressions for potential of a line charge between two parallel grounded conductors - method of images and series solution
I am trying to figure out how two expressions for the potential of a line charge between two grounded parallel conductors are equivalent. Showing the equivalence of the two expressions seems more ...
-1
votes
2
answers
54
views
How to find a closed-form expression for the sequence $x_n = x_{n-1}(1.1) + 100$
I have the sequence $\mathbf{x_n = x_{n-1}(1.1) + 100}$ with $\mathbf{x_0 = 100}$.
How can I calculate $\mathbf{x_n}$ as an explicit function of $\mathbf{n}$?
- 35
1
vote
1
answer
42
views
Why this sequence converges to the intersection?
Consider the following set up (you can follow on desmos):
In a plane we have to circles that intersect at 2 point, circle $A$ and circle $B$, with centers at $A$ and $B$ respectively. For simplicity ...
- 13
0
votes
1
answer
64
views
0
votes
1
answer
79
views
Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$
Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$
I'm quite sure that this statement is true, but I'm having a little difficulty ...
- 101
1
vote
1
answer
32
views
Confusing notation of sequences in $k$-dimensional Euclidean spaces.
Suppose $(x^{(n)})$ is a sequence in $\mathbb R^k$, $k \in \mathbb N$. From what I understand, $(x^{(n)})$ is a sequence of sequences $(x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, ..., x_k^{(n)})$. Or in other ...
- 187
0
votes
2
answers
40
views
Trouble understanding difference between the following two results involving the number of terms in the set $\{n: s_n \gt \lim \sup s_n\}$
There are two results written in Ross' book on Elementary Analysis, which state the following -
If $L = \limsup s_n \neq \infty$, then for every $\alpha > L$, the set $\{n: s_n \gt \alpha\}$ is ...
- 187
0
votes
1
answer
33
views
How to study the convergence of a piecewise serie?
I try to study the convergence of the series of the general term:
$$U_n=\begin{cases}
\frac1n ,\text{ if $n$ is square}\\
\frac1{n^2},\text{ else }
\end{cases}$$
But I didn't find a way to start, even ...
- 29
21
votes
5
answers
468
views
Interesting infinite product $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$
I have found an interesting family of infinite products. The most interesting one of them being:
$\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$...
- 717
-1
votes
0
answers
32
views
Sequence or probability problem [closed]
What way should be continued the sequence 1 2 3 4 5 6 7 8 9 10?
My answer is the sequence above representss natural numbers sequence. The sequence is bounded below (1) and does not have upper limit, ...
- 27
6
votes
2
answers
120
views
Distribution of $Y = X \bmod 2\pi$ with $X$ being a Cauchy distribution
Let $X$ be a Cauchy distribution with parameter $\theta$, that is to say, its density function is:
$$
f(x;\theta) = \frac{\theta}{\pi(x^2 + \theta^2)}
$$
I'm asked to get the distribution of $Y$ where ...
-3
votes
0
answers
45
views
Is this sequence arithmetico geometric? [closed]
I am im desperate need of help to solve a mathematic exercise.
I have to determine if this sequence is arithmetico geometric but I struggle explaining it.. I am supposed to present it in front of my ...
- 1
0
votes
1
answer
76
views
Show $\epsilon[\frac{x}{\epsilon}] \rightarrow x$ when $\epsilon \rightarrow 0$
The question is to show that $\epsilon[\frac{x}{\epsilon}] \rightarrow x$ for $\epsilon \rightarrow 0$.
Here, $x \in \mathbb{R}^n$ and $\epsilon \in \mathbb{R}$.
Intuitively, it does not make sense to ...
- 160
-3
votes
0
answers
50
views
Does this function go towards $0$? [closed]
Consider the function defined on positive integers by the formula $f(n)=div(n)/n$, where $div(n)$ is the number of positive integer divisors of $n$. Does that function go toward $0$ as $n$ goes to $\...
- 18.5k
0
votes
0
answers
21
views
Questions about Complex infinite Products Convergence Implication
Actually there is a theorem in Infinite Products states that
$\textbf{$\prod_{n=1}^{\infty}u_n$ is convergent if and only if there is an $m \in \mathbb{Z^+}$ so that $\sum_{n=m}^{\infty} \log{u_n}$ is ...
0
votes
0
answers
27
views
Problem in proof of theorem involving $\lim \sup$ and $\lim \inf$
Source: Elementary Analysis By Kenneth A. Ross
Theorem: If $s_n$ converges to a positive real number $s$ and $(t_n)$ is any sequence, then
$$\lim \sup s_nt_n = s\cdot\lim \sup t_n$$
Proof: We first ...
- 187
5
votes
3
answers
138
views
Computation of $\displaystyle{\sum_{n=1}^{\infty}\frac{\sin nx \cdot \sin ny}{n^2}}$
First I used the identity
$$\sin nx \cdot \sin ny=\cos(n(x-y))-\cos(n(x+y))$$
and the sum turned into the following
$$\sum_{n=1}^{\infty}\frac{\sin nx \cdot \sin ny}{n^2}=\sum_{n=1}^{\infty}\frac{\cos(...
1
vote
0
answers
181
views
Some questions about a sequence and concepts i developed to find its nth term.
So I developed these concepts and they seem to be useful to find nth term of certain types of sequences. I have only given definitions and rules and other things, i decided not to write what led me to ...
- 29
-2
votes
2
answers
88
views
Using the AM-GM inequality, show that,for $n ≥ 1$, $\left(1+\frac{1}{n}\right)^n < e < \left(1+\frac{1}{n}\right)^{n+1}$ [closed]
I have been struggling in solving the following question:
Using the AM-GM inequality, show that, for $n ≥ 1$, $$\left(1+\frac{1}{n}\right)^n < e < \left(1+\frac{1}{n}\right)^{n+1}$$
0
votes
0
answers
23
views
Help with understanding a proof involving subsequences, $\lim \sup$ and $\lim \inf$
Source: Elementary Analysis By Kenneth A. Ross
I am having trouble understanding the last part of the proof of theorem 11.7.
"Theorem 11.2(i) shows that a monotonic subsequence of $s_n$ ...
- 187
0
votes
0
answers
55
views
How do I get the ratio when only given 3 sums of a geometric sequence
The exercise:
The sum of the first four terms of a geometric progression is $520$. The sum of the first five terms is $844$. The sum of the first six terms is $1330$. Find the common ratio of the ...
- 1
0
votes
1
answer
50
views
At which value of K, the sum converges
I have a infinite sum that looks like the following:
$$ \sum_{k = 0}^{\infty} \left(\frac{1}{1 + k/A}\right)^a \frac{(-i B)^k}{k!} $$
Here, A is a positive real number and $a$ is a positive integer. ...
- 67
0
votes
3
answers
67
views
Find least $\lambda$ for which recursive sequence is always positive
Find the least $\lambda$ for which the sequence $\{b_n\}$ defined by $b_1=1$, $b_2=\lambda-1$ and $b_{n+2}=\lambda(b_{n+1}-b_n)$ is always positive.
I guess $\lambda=4$, which yields $b_n=2^{n-1}(n+1)...
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