For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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1
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2answers
31 views

Is there any formula to find the nth element in a sequence where common difference (d) is varying with a constant rate?

To explain my question, here is an example. Below is an AP: 2, 6, 10, 14....n Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is ...
0
votes
3answers
51 views

Write an expression in terms of $n$ for the $n$th term in the following sequence: $9,16,25,36,49$

Write an expression in terms of $n$ for the $n$th term in the following sequence: $9,16,25,36,49$ The difference is $+ 7 , + 9 , + 11 , + 13, + 15 , + 17 ,$ etc The difference is not constant so ...
1
vote
2answers
40 views

How can I find the limit of the $s_n=n[1+(-1)^n]$

$s_n=n[1+(-1)^n]$ Any hints on how to get started with this one?
7
votes
2answers
150 views

I derived a new formula related to arithmetic sequences, I think!

First of all, I am a 12th student so I don't know how to write research notes. So please forgive me if my writing is not so impressive! I don't know what to do to let the world know about whatever I ...
0
votes
0answers
18 views

Alternating sum of polygamma functions

I've come across the following sum while doing a problem: $$\sum_{n=0}^\infty (-1)^n \psi^{(1)}(m+np)$$ Where $m \in (0,\infty)$ and $p \in \mathbb{N}$ and $\psi^{(1)}$ denotes the first derivative ...
3
votes
1answer
66 views

Does $a_n \sim b_n$ imply $\sum_n a_n \sim \sum_n b_n$ for $a_n, b_n>0$?

I am almost embarrassed to asked to this question, but after considering it for a while I realize I need some help. In the following $a_n, b_n >0$. So, by limit comparison test if $a_n = O(b(n))$ ...
16
votes
3answers
301 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial coefficient ...
7
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2answers
122 views
+100

$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.

Let $m \ge 2 -$ fixed positive integer. The sequence of non-negative real numbers $\{x_n\}_{n=1}^{\infty}$ is that for all $n\in \mathbb N$ $$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$$ ...
0
votes
0answers
23 views

More elegant way of expressing Lagrange polynomial.

On wikipedia the Lagrange polynomial looks messy, I think I found a elegant way to express the Lagrange polynomial: Like this (where $\Delta L(x)$ represents $L(x+1)-L(x)$: $$L(x)=\sum_{i=0}^{\infty}...
0
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1answer
98 views
+50

Express the sequence which alternates three negative ones and three positive ones

I asked the same question here:Three negative and three positive $1$s in a serie But when I see that it was closed because of being unclear I decided to make It better and ask again. First I want to ...
0
votes
0answers
34 views

Limit involving cumulative binomial distribution function

I want to find limit of the expression below as $n\to\infty$(If the limit is zero, then I will use it) $$\frac{1}{(p+1)^{2n}}\sum\limits_{k=0}^{2n}\binom{2n}{k} p^{2n-k}\frac{|n-k|}{\sqrt{n}}\qquad \...
2
votes
1answer
33 views

Series Converges Pointwise but not Uniformly

Consider $\sum_{n=1}^{\infty}\frac{x^2}{n(n+x^2)}$ on $[0,\infty)$. To show pointwise convergence, I used $\frac{x^2}{n(n+x^2)}\leq\frac{x^2}{n^2}$ on $[0,\infty)\implies\sum_{n=1}^{\infty}\frac{x^2}{...
2
votes
2answers
28 views

Pattern with the the titration of summations.

While dealing with a question with finding an explicit form for a sequence I noticed something: $$\sum_{x_0=0}^{n-1} 1=\frac{n}{1!}$$ $$\sum_{x_0=0}^{n-1} \sum_{x_1=0}^{x_0-1} 1=\frac{n(n-1)}{2!}$$ ...
-5
votes
1answer
119 views

Telescoping function Revealed.

Part B: I found Summation $$\sum_{n=0}^\infty\frac{x^n \ (-1)^n}{(y+1)^{n+1}} = \frac{1}{(x+y+1)}$$ However $x$ is related to $y$. $y \ge |x|$. You may derive the result and present it by ...
1
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1answer
40 views

Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression

I'm thinking we could do a contradiction, maybe showing that one of the primes is a composite number if they are in a sequence, but I'm not sure how to finish this up. I had this as a math problem in ...
1
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2answers
39 views

To prove the given inequality

Question:- If $a,b,c$ are positive real numbers which are in H.P. show that $$\dfrac{a+b}{2a-b} + \dfrac{c+b}{2c-b} \ge 4$$ Attempt at a solution:- I tried it by AM-GM inequality, but got stuck ...
1
vote
1answer
68 views

Closed form of the series.

Write the following series in closed form $$\frac{1}{a-1}+\frac{2}{a-2}+\frac{2}{a-3}+\frac{1}{a-4}+\frac{1}{a-5}+\frac{2}{a-6}+\frac{2}{a-7}+\frac{1}{a-8}+\frac{1}{a-9}+\frac{2}{a-10}+\frac{2}{a-11}...
-3
votes
1answer
49 views

To find product of the given sequence [on hold]

To find the product of the given sequence $$(a+1)(a+3)(a+5)\dot \ ... \dot\ (a+2n-1)$$ where $0<a<1 $.
2
votes
3answers
103 views

Prove that $\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$

Prove that $$\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$$ My idea is to find the Taylor series of $\frac{1}{(e^x-1)^2}$, but it seems not useful. Any helps, thanks
37
votes
25answers
34k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is know as "The sum of the first $n$ positive ...
0
votes
2answers
25 views

Does a positive sequence $x_n$ whose limit is zero achieves suprimum achieves at finite value of $n$?

I am working on important question which is useful for my work. Say suppose we have a positive sequence whose limit is zero. Then I believe that supremum of $x_n$ over $n$ is achieved at some finite ...
0
votes
1answer
584 views

Does the series $\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$ converge?

Does the series $$\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$$ This is supposed to be an alternating series but I can't seem to figure out what the $b_n$ is in this case. is there some ...
-5
votes
3answers
131 views

Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$, and Integral of $e^{iPi}$ has derivative = 1. [on hold]

It is a simple question: Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2, and the integral of $-e^{i*Pi}$, has derivative =1. Think Kepler's third law is a constant. [an ...
-4
votes
1answer
93 views

new equation for $\int_0^ t e^{-x2} dx$? [closed]

fact! $$\int_0^ x e^{-x^2} dx$$ $$=e^{-x^2}\sum_{n=0}\frac{(2^n)x^{2n+1}}{{(2n+1)!!}}$$ Well the equation was new to me, when I derived by shear integration, and that is a cold HARD fact.
3
votes
1answer
188 views

help verifying equation $\int_0^ x \frac{1}{1+t^n} dt$

As a follow up to a previous posting addressing the integral of $1/ (t^n+1)$ for $n\in \Bbb{N}$ I found the following $$\int_0^ x \frac{1}{1+t^n}\, dt=\sum_{i=0}^{\infty}\frac{(i!)(n^i)x^{in+1}} {(x^...
1
vote
1answer
73 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
2
votes
1answer
67 views

Explaining a Series Expansion for the Divisor Function

The "sum-of-divisors" function, defined as $\sigma(n)=\sum_{d\,\mid\,n}d$, can be expressed as $$\sigma(n)=\sum_{i=1}^n\sum_{j=1}^i\cos\Big(2\pi n \frac{j}{i}\Big)$$This expansion makes logical sense. ...
22
votes
1answer
592 views

Is there any other way to prove this fact? (non-existence of slowest diverging series)

Let $a_n>0$ and $S_n=\sum_{k=1}^{n}a_n$. If $\lim_{n \rightarrow \infty}S_n = +\infty$, then $\sum_{n=1}^{\infty}\frac{a_n}{S_n}=+\infty$. I think this is an important example because it tells us ...
0
votes
0answers
19 views

Prove $X_n \nrightarrow X = \bigcup_{k=1}^{\infty} \{|X_n - X| > \frac{1}{k}\}$ [on hold]

Probability with Martingales: Important inequalities: 1, 2 $$\liminf x_n > z \to \liminf(x_n > z)$$ $$\liminf x_n < z \to \limsup(x_n < z)$$ What I tried: I think the ...
0
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0answers
15 views

Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
4
votes
1answer
83 views

Any hints on how to prove that $\ln{1\over 2\sin\left({90\over \pi}\right)}=\sum_{n=1}^{\infty}{(-1)^{n-1}B_{2n}\over 2n(2n)!}$?

How do you prove that $$ \ln\left(1 \over 2\sin\left(1/2\right)\right) = \sum_{n = 1}^{\infty}{\left(-1\right)^{n - 1}\,B_{2n} \over 2n\left(2n\right)!}\ ?\tag1 $$ where $B_{2n}$ is a Bernoulli ...
3
votes
3answers
93 views

Recurrence relation solution $a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}}$

I want to find the analytic form of the recurrence relation $$a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}},\,a_{1}=2,\,a_{2}=1 $$ but when looking at the results they seem chaotic. Is it possible that it ...
8
votes
1answer
97 views

How to solve such problem based on series?

I was given this problem on series by a friend. If $$y=\sqrt{4 + \sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\ldots}}}}$$ then how to solve such problem. I don’t want the full answer, rather, insights, ...
-1
votes
0answers
32 views

Taylor Series coefficients

I have performed a Taylor Series expansion of a 2-D function in variables (y1,y2) and got something like: $f(y_1,y_2) = 3y_2 + 0.5y_1^2 + ...$ My question is that I would like this to "match" to ...
-1
votes
1answer
38 views

Checking if $(x^{2n}-1.5x^n+\frac12)^\infty_{n=1}$ converges uniformly on$ [\frac12,1]$ and on $[0,\frac12]$

Checking if $\left(x^{2n}-1.5x^n+\frac12\right)^\infty_{n=1}$ on $ [\frac12,1]$ and on $[0,\frac12]$. What tests/techniques do I use to prove uniform convergence?
2
votes
2answers
59 views

Growth of the set $\lbrace 2^n+3^m \rbrace$ in the integers

Let $A=\lbrace 2^n+3^m \ | \ n,m\in{\mathbb N}\rbrace$, and denote by $a_k$ the $k$-th element of $A$ in order from smallest to greatest ; thus $a_1=2,a_2=3,a_3=4,a_4=5,a_5=7\ldots$. Is it known ...
1
vote
1answer
23 views

Is there any property of the sigma notation through which we could find 'sigma(i=1 to n) of $i^4$'?

The way we came up with the formulae of similar types like - the summation of cubes of the first 'n' natural numbers proved to be a cumbersome one while trying the above one.
1
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4answers
80 views

Finding the finite product $\prod_{k=2}^{\infty}\frac{k^2}{k^2-1}$.

Let $\displaystyle\prod_{k=2}^{\infty}\frac{k^2}{k^2-1}=P^2$. I need hint to find $P.$
-2
votes
2answers
80 views

Finding the sum $\frac{5}{12}+\frac{5}{36}+\frac{5}{72}+…+\frac{5}{5580}.$ [on hold]

I need help to find a rule for the sum $$\frac{5}{12}+\frac{5}{36}+\frac{5}{72}+...+\frac{5}{5580}.$$
1
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2answers
35 views

Is it possible to find the partial sum of a series if I know the infinite sum?

Say, I know the value of $\displaystyle\sum_{r=0}^{\infty}t_r$ (assuming it is convergent) and I want to find out the $n$th partial sum $\displaystyle\sum_{r=0}^{n}t_r$ for the sequence $\{t_r\}$ from ...
2
votes
1answer
40 views

Relation between these two series

Assume a constant $\alpha$ and $N$ positive integers $\{n_1,...,n_N\}$. What is the relation between $\frac{N\alpha}{\sum_i{n_{i}}}$ and $\sum_i{\frac{\alpha}{n_{i}}}$ when $N\rightarrow\infty$? $N$ ...
1
vote
0answers
15 views

Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
0
votes
2answers
35 views

Partial summation formula for $\sum_{k=1}^{\infty}\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}}$.

I am stuck in finding the partial summation formula for the following series. I need hint. $$\sum_{k=1}^{\infty}\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}}$$
2
votes
0answers
33 views

Asymptotic vlaue of $ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $

Inspired by this question I tried to find an asymptotic formula for $$ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $$ With the observation: $$ f(n)=\sum_{i=0}^n\frac{\lfloor \sqrt{i}\rfloor+\...
0
votes
1answer
24 views

Expressing Tornheim sums in terms of Riemann's Zeta

If $$T(a,b,c)=\sum_{r\geq1}\sum_{s\geq1} \frac{1}{r^as^b(r+s)^c}$$ How to prove that : $$T(3,1,2)=-\frac13 \zeta(6)+\frac{\zeta^2(3)}{2}$$ I tried some algebraic manipulations but did not work. Can ...
0
votes
2answers
49 views

Existence of sequences of natural numbers

Let $x \in [0, 1)$ does there exists a sequence of natural numbers $a_n, b_n$ such that the sequence $a_n/b_n$ converges to $x$.
1
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0answers
43 views

Check whether my solution holds good or not.

A sequence $\{x_n\}$ be such that every subsequence of it has a further subsequence converging to $1$ then show that the main sequence will converge to $1$. My work : Let us try to prove it by the ...
1
vote
2answers
38 views

Why does changing the center of a geometric power series change the interval of convergence?

I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{...
2
votes
3answers
34 views

Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, Show that $y_n → 0$.

Let $(x_n)$ be a sequence. Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, i.e. $\sigma$ is a permutation of the set of natural numbers. Show that $y_n ...