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

learn more… | top users | synonyms (5)

0
votes
0answers
39 views

CEOI 1994 task Is there a sequence of n number where each sum of any p consecutive elements is positive and sum of any q is negative

Write a program which reads three positive integers $n, p, q.$ Decide whether or not there exists a sequence of n integers such that the sum of any p consecutive elements is positive and the sum of ...
10
votes
3answers
68k views

Whats infinity divided by infinity?

This should be a simple question but i just want to make sure. I know from infinity/infinity is undefined. However if we have 2 equal infinities divided by each other it would be 1? And if we have ...
0
votes
1answer
22 views

How many terms lies between

If a have a series from $n^2$ to $(2n)^2$, i.e. $n^2+(n+1)^{2} +...+ (2n)^2$, how many terms lie between $n^2$ and $(2n)^2$ ? Is it $n+1$ terms, or is it $n$ terms, and how to prove that.
2
votes
3answers
63 views

Convergence and Limit of a Sequence

I am looking at the sequence and I'm trying to see what happens as $n\to\infty$ $$ a_n = \frac{\prod_{1}^{n}(2n-1)}{(2n)^n} = \frac{1\cdot3\cdot5\cdot...\cdot(2n-1)}{(2n)^n} $$ By inspection, I can ...
1
vote
3answers
198 views
3
votes
4answers
883 views

Maclaurin series for $\frac{x}{e^x-1}$

Maclaurin series for $$\frac{x}{e^x-1}$$ The answer is $$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$ How can i get that answer?
1
vote
3answers
149 views

Consider the sequence ${x_n}$ defined by $x_n = [nx]/ n$

Consider the sequence ${x_n}$ defined by $x_n = [nx]/ n$ for $x\in\mathbb R$ where $[·]$ denotes the integer part. Then ${x_n}$ (a) converges to $x$. (b) converges but not to $x$. (c) does not ...
3
votes
5answers
131 views

How to find the MacLaurin series of $\frac{1}{1+e^x}$

Mathcad software gives me the answer as: $$ \frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$ I have no idea how it found that and i don't understand. What i did is ...
5
votes
5answers
7k views

Famous puzzle: Girl/Boy proportion problem (Sum of infinite series)

Puzzle In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is ...
1
vote
1answer
33 views

Transforming a sequence to distinguish a limit

This might be the wrong place to ask this question, but I figured I might get some creative answers: I have a decreasing sequence $\{a_n\}_{n \geq 1}$ with $a_k \in (0,1)$ for all $k$ and $a_n \to ...
4
votes
4answers
322 views

Evaluate $ \sum_{n=1}^{\infty} \frac{\sin \ n}{ n } $ using the fourier series

I am a beginner with Fourier series and I have to evaluate the sum $$\sum_{n =1}^{\infty}{\sin\left(n\right) \over n}$$ I don't know which function I have to take to evaluate the fourier series ...
8
votes
4answers
378 views

Convergence of $\sum_{k=1}^{n} f(k) - \int_{1}^{n} f(x) dx$

I had asked this question sometime ago here. Now I have a question which I think is related to it. Let $f$ be an increasing function (continuous of course!) with $f(1)=0$. Consider the sequence ...
0
votes
1answer
35 views

What does this point about triangular number mean

I was reading about triangular numbers from Wikipedia. I makes following point on the above web page: The number of line segments between closest pairs of dots in the triangle can be represented ...
8
votes
3answers
920 views

$1 + 1 + 1 +\cdots = -\frac{1}{2}$

The formal series $$ \sum_{n=1}^\infty 1 = 1+1+1+\dots=-\frac{1}{2} $$ comes from the analytical continuation of the Riemann zeta function $\zeta (s)$ at $s=0$ and it is used in String Theory. I am ...
2
votes
2answers
150 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
2
votes
0answers
47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
5
votes
3answers
73 views

Is $\sum_{n=1}^\infty \frac{m}{(n+m)^2}$ bounded for all $m\in\mathbb{N}$?

I'm trying to figure out if there is a finite constant $C$ such that $\sum_{n=1}^\infty \frac{m}{(n+m)^2}\leq C$ for all $m\in\mathbb{N}$. I can see that $\sum_{n=1}^\infty ...
0
votes
1answer
16 views

Sequence m(k) with $\frac{m(k)}{3\cdot k\cdot log(k)}>0$ for $k\rightarrow\infty$

I'm looking for a sequence m(k) which fullfills the condition $\frac{m(k)}{3\cdot k\cdot log(k)}>0$ for $k\rightarrow\infty$. log(k) means the natural logarithm and m,k are positive integers. ...
0
votes
1answer
34 views

How can we solve a multi-variable recurrence relation in closed form, when the number of terms is also variable?

Consider the formula $f(x, y) = f(x, y-1) + 2 \sum\limits_{i=1}^{x-1} f(i, y-1) $ The factor '2' makes this not expressible cleanly as $f(x, y) = f(x, y-1) + f(x-1, y)$, which is solved here using ...
1
vote
2answers
34 views

Engineering Mathematics Problem with Taylor's Series

This is a problem from Engineering Mathematics book by K.A. Stroud 7th edition, Exercise 18, Chapter 12 Further problems. It has been given in a physics manner, but it just requires manipulation of ...
0
votes
1answer
23 views

Fourier Series Relation Time - Frequency

I want to study and understand the relation between time and frequency with the help of Fourier Series. Can you indicate me some papers, or some example?
0
votes
2answers
22 views

Accumulation/limit points

The problem is: For the sequences $(-1)^{n+1}\dfrac{n}{2n+5}$, $(-1)^{n+1}\dfrac{n}{2n^2+5}$ and $\dfrac{1}{\sqrt{n}}\cos\Bigl(1+\dfrac{1}{n}\Bigr)$ determine the accumulation ...
2
votes
0answers
38 views

Expansion of Weierstrass elliptic function in second period

I would very much like to find expansions of the Weierstrass $\wp$ and $\zeta$-functions for small absolute values of the second period $\omega_2$. So, more precisely, I would like, for ...
5
votes
4answers
237 views

What is the infimum?

I'm trying to find $\inf_{n \in \mathbb N} (\sin(n))^2 $. I think that the answer is $0$ but I couldn't prove it. I appreciate any help.
1
vote
2answers
58 views

How to interpret $\sum_{n\in \mathbb N^{d}} \frac{1}{n^{p}};$ and when it is converges?

I know that: $\sum_{n\in \mathbb N} \frac{1 }{n^{p}}$ converges if $p>1$ and diverges if $p\leq 1$ My Question is: What is an analogue this in more than one variable (say $d$)? Does it make ...
0
votes
0answers
28 views

step by step solution of $y''+(x-1)y=0$ by Frobenius method

I've tried solving this equation, $y''+(x-1)y=0$, but honestly I'm not sure if I've done it right. Using Frobenius Method, I got $r_1=1$ and $r_2=0$ as the indicial roots which I think under case 3 as ...
15
votes
1answer
296 views

About sequences of positive integers

Prove there is no sequence of positive integers $(x_n)_{n \ge 1}$ so that: $$ x_{n+2} = x_{n+1} + x_{x_n} \quad \forall n\ge1 $$ I think the idea is to find two different values for the same index.
-4
votes
1answer
38 views
-4
votes
0answers
66 views

What inconsistencies/ fallacies do techniques used in assigning -1/12 to the infinite sum “1+2+3+4+…” contain? [closed]

It would be helpful if the downvoters could communicate the reason for downvoting the question. I think I have clarified the question adequately below. If I am missing something, requesting you to ...
2
votes
4answers
74 views

$ {a_n}^{1/k} \rightarrow m^{1/k}$

Let $k$ be a fixed positive natural number and $\{a_n\}$ be a sequence of positive reals converging to $m$ then show that $$(a_n)^{1/k} \hspace{.1cm} \text{converges to } \hspace{.1cm} m^{1/k}$$ as ...
1
vote
1answer
22 views

Divergence of sequence contradict [closed]

If two sequence diverge there sums or difference can either converge or diverge ?can anyone tell me am I right or wrong .
7
votes
4answers
436 views

Trying to show $1-\frac12 -\frac {1} {4}+\frac {1} {3}-\frac {1} {6}-\frac {1} {8}+\frac {1} {5}-\cdots =\frac {1} {2}\log 2$

Now I think the lhs can be rewritten as $$\sum _{n=1}^\infty \left( \dfrac {1} {2n-1}-\dfrac {1} {2\cdot 3^{n-1}}-\dfrac {1} {2^{n+1}}\right) =\dfrac {1} {2}\log 2$$ I guess one way to do this may be ...
-1
votes
0answers
108 views

Is $1+2+3+4+\cdots=-\frac 1{12}$ true? [duplicate]

Hello (it's my first post here!), I have a strange question. I heard that (under certain conditions): $$ 1+2+3+4+\ldots=\sum_{k=1}^{\infty}k=-\frac{1}{12} $$ Is it REALLY true? And - if yes - how to ...
1
vote
1answer
33 views

$\{na_{n}\} \subset \ell^{\infty}(\mathbb Z)$ if $a_{n}\in \ell^{1}$?

Suppose $\{a_{n}\} \subset \ell^{1}(\mathbb Z)$ (Sequence space). MY Question: Can we expect $\{na_{n}\} \subset \ell^{\infty}(\mathbb Z)$?
0
votes
1answer
44 views

Is there a theorem relating sequences to its series or vice versa?

I only have these in mind Theorem: If a series $\sum_n a_n$ of real numbers converges then $\lim_\limits{n \to \infty} |a_n|=0$ Divergence test: If $\lim_\limits{n \to \infty} a_n \neq 0$, then the ...
3
votes
1answer
74 views

Finding a power series representation for $\left(\frac{x}{2-x}\right)^3$

Find a power series representation for $\displaystyle\left(\frac{x}{2-x}\right)^3$ My approach is in finding something similar to $\displaystyle\left(\frac{x}{2-x}\right)^3$ to which I can easily ...
0
votes
1answer
39 views

Finding a positive lower bound of hte sequence $\frac{\sqrt[n]{n!}}n$

I am given a sequence {(n'th root of n!)/n}. Can I show that the sequence is bounded below by a real no. which is greater than 0, by not calculating the limit of it....???thank you
1
vote
2answers
117 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
0
votes
1answer
53 views

Proof of a continuous function?

I would like to know how tho prove or disprove the following: Prove the following statement: Every continuous function $f:[a,b] \mapsto \mathbb{R}$ (with $a < b$) is (from above) bounded. I have ...
2
votes
2answers
64 views

Given $x_0>0$ and $a>0$, if $x_{n+1}=\frac{2a^2x_n}{x_n^2 + a^2}$ for every $n\ge0$ then $\lim\limits_{n\to\infty}x_n=a$

Given $x_n$ be a sequence of positive numbers such that $$x_{n+1}=\frac{2a^2x_n}{x_n^2 + a^2} , a>0.$$ Show that $$\lim_{n\to\infty}x_n=a.$$ I thought of first proving sequence bounded and ...
2
votes
0answers
53 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
1
vote
1answer
84 views

Proving a series of functions does not converge uniformly on $\mathbb{R}$

I'm working through a question (it's a question with parts that builds on each other) that overall will show: a series of functions does not converge uniformly on $\mathbb{R}$ by showing the ...
1
vote
1answer
29 views

Find Closed-form expression of a integer sequence.

We have a integer sequence $u_{n+1}=2p.u_{n}-u_{n-1}$ (p is a positive integer, $n\geq 3$ ) When $u_{1}=1; u_{2}=p$ then with a regular way I can find Closed-form expression of the sequence. ...
1
vote
1answer
32 views

can you consider a series to be a sequence of sums?

for example sequence: $1/(2^n), \qquad n\ge 0$ sequence for the series: $1, 1.5, 1.75, 1.875, \ldots$ and if so, does that mean you can use/extend sequence theorems for series?
3
votes
1answer
130 views

Hypergeometric Function simple identity

I must prove this property but I really have no idea of how to prove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems it's a 'simple' property, but I haven't been able to ...
2
votes
1answer
32 views

Correctness of proof for the convergence of a series

Does the following series converge? $\sum_{n=1}^\infty \frac{5^n + (-1)^n}{2^n3^n}$ What I've done so far. $0\leq \sum_{n=1}^\infty \frac{5^n + (-1)^n}{2^n3^n} \leq \sum_{n=1}^\infty \frac{5^n + ...
0
votes
1answer
47 views

Why does the limit not change in the given summation series even after substituting $p=-n$ in the given question?

$$ \begin{align} DFS[x^*(-n)] &= \frac{1}{N}\sum^{N-1}_{n=0}x^*(-n)e^{-j2\pi kn/N}\\ &= \left[\frac{1}{N}\sum^{N-1}_{n=0}x(-n)e^{j2\pi kn/N}\right]^*\\ &= ...
1
vote
3answers
183 views

How to calculate the limit of this sum with different methods? [duplicate]

It's a basic question , but what are the common methods to calculate limits like this one: $$\sum_{k=1}^\infty \frac{3k}{7^{k-1}}$$
2
votes
6answers
127 views

How to prove $\lim_{n\rightarrow\infty}nx^n=0$ without L'Hôpital's rule, where $x \in [0,1)$??

How to prove $$\lim_{n\rightarrow\infty}nx^n=0$$ without L'Hôpital's rule? where $x \in [0,1)$ and $n=1,2,3,...$. I know one of way to prove this is to treat $n$ is real, and $n$ and ...
3
votes
3answers
79 views

Prove that the sequence $a_n=\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$ is monotonically decreasing sequence

Prove that the sequence $$a_n=\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$$ is monotonically decreasing sequence. I tried $a_{n+1} - a_{n} < 0$, but i was not able to do it. Help ...