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

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0
votes
1answer
15 views

Operator for comparing an n-tuple

Suppose you have to compare the following two finite ordered list of elements (tuples): $(\psi_{i}, R_{i}, A_{i}, \eta_{i})$ and $(\psi_{i}^{*}, R_{i}, A_{i}, \eta_{i})$ and for instance it turns out ...
3
votes
2answers
36 views

Sum of a Finite Sequence of Terms:$18, 25, 32, 39, … ,67$

Ok I know this question maybe too easy. What is the sum of a finite sequence of terms? $$18, 25, 32, 39, ... ,67$$ The answer is $340$. I use the formula: $${ S = \frac{n}{2} \times (a_1 + a_n) ...
6
votes
3answers
71 views

sequence of primes in arithmetic progression

The question is: Suppose $p_1<p_2<...<p_{15}$ is a sequence of prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$. Let ...
2
votes
1answer
58 views

How to find the Summation S

Given function $f(x)=\frac{9x}{9x+3}$. Find S: $$ S=f\left(\frac{1}{2010}\right)+f\left(\frac{2}{2010}\right)+f\left(\frac{3}{2010}\right)+\ldots+f\left(\frac{2009}{2010}\right) $$
1
vote
1answer
103 views

Is there a pattern of the length between one even Fibonacci number and another?

I had seen a math problem asking for the sum of all even Fibonacci numbers up to 4 million, but I still need to know this: Is there an obvious pattern of the distance between a even Fibonacci number ...
1
vote
0answers
70 views

A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot ...
5
votes
4answers
270 views

Check whether $\sum\limits_{n=1}^{\infty}\frac{z^n}{(1-z^n)^k}=\sum\limits_{n=1}^{\infty}\sigma_{k-1}(n)z^n$

Is it true that $$\sum_{n=1}^{\infty}\frac{z^n}{(1-z^n)^k}=\sum_{n=1}^{\infty}\sigma_{k-1}(n)z^n$$ If yes, how can I prove it?
0
votes
3answers
3k views

Bounded monotonic sequences

I'm having difficulties understanding how to show what sequences are monotonic and/or bounded. I know that a bounded monotonic sequence converges, but what about a sequence that is just monotonic or ...
12
votes
4answers
2k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
0
votes
1answer
27 views

Geometric Progression of Air removed by an Air Pump

If one third of the air in the tank is removed by each stroke of an air pump, what fractional part of the total air is removed in 6 strokes Answer is 0.0877 I was thinking this was some sort of ...
2
votes
1answer
83 views

Range of inner product of a sequence and its permutation

$a^n :=(a_i)_1^n$ is a finite sequence of real numbers of length $n$, where $\sum\limits_{i=1}^n a_i=0$ and $\sum\limits_{i=1}^n a_i^2=1$. Consider $s_n(a^n,\sigma):=\sum\limits_{i=1}^n ...
0
votes
0answers
46 views

A divergent sequence of integrals

Let {$s_{n}$} be a sequence of increasing step functions that converges pointwise to a limit function $f$ on an interval $I$ that is unbounded. $f(x)\ge 1$ almost everywhere on $I$ . ...
1
vote
0answers
18 views

Sum of Bell Polynomials of the Second Kind

A problem of interest that has come up for me recently is solving the following $$\frac{d^{n}}{dt^{n}}e^{g(t)}$$ There is a formula for a general $n$-th order derivative of a composition as shown ...
1
vote
2answers
51 views

Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
2
votes
1answer
50 views

Prove that $a_j=b_j$ for the $2$ sequences $a$ and $b$

Let $n$ is a natural number and $(n,6)=1$. Given $2$ sequences $a$ and $b$ such that $a_1>a_2>\ldots a_n$ and $b_1>b_2>\ldots b_n$. And for all $1 \leq j < k <l \leq n$, it is ...
2
votes
4answers
383 views

Sum function of a series

Does anyone know what is the sum function $f(x)$ of the series $\displaystyle\sum_{n=1}^\infty \frac{\cos(nx)}{n^2}$? I have no idea how to find a sum function... Any help would be appreciated.
3
votes
2answers
35 views

Limit and limit points

What is the basic difference between limit and limit points, and if a sequence has one unique limit how it can have a lot of limit points
8
votes
4answers
964 views

Does the series $\sum\limits_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$$ As $$\frac{1}{n\sqrt[n]{n}}=\frac{1}{n^{1+\frac{1}{n}}},$$ I was thinking that you may consider this as a p-series ...
2
votes
2answers
202 views

Find sum of $n$ terms of the series $12+14+24+58+164+\cdots$

Find sum of $n$ terms: $12+14+24+58+164+\cdots$ I have tried my best but could not proceed
9
votes
1answer
278 views

If $m\geq2$ is an integer, then $\sum\limits_{n=1}^{\infty}m^{-n^2}$ is irrational

Let $m \geq2$ be an integer. I want to ask how to prove that the sum of the following series is irrational: $$\sum _{n=1}^{\infty} \frac{1}{m^{n^2}}$$
1
vote
1answer
32 views

Alternating Series Test Conditions

I'm learning about the Alternating Series Test which can be found here. My question is: Can you give an example of where $\lim_{n\to\infty} b_{n} = 0$ but $b_{n}$ is an increasing sequence?
11
votes
1answer
252 views

If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum a_n/s_n$ diverges as well

So I've been trying to figure out how to prove the following. Let $(a_n)$ be a sequence of positive numbers such that $\sum\limits_{n=1}^\infty a_n =\infty$, and define $s_n=\sum\limits_{i=1}^n ...
19
votes
5answers
1k views

Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$

Today, we had a math class, where we had to show, that $a_{100} > 14$ for $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$ Apart from this task, I asked myself: Is there a closed form for this ...
-2
votes
1answer
268 views

Iterating $(a,b)\mapsto(a+b+\sqrt{a^2+b^2} ,a+b-\sqrt{a^2+b^2})$

Assume that $a_0=-2$, $b_0=1$, and that, for every $n\ge0$, $$a_{n+1}=a_n+b_n+\sqrt{a^2_n+b^2_n} \qquad b_{n+1}=a_n+b_n-\sqrt{a^2_n+b^2_n}$$ How to find $a_{2012}$?
9
votes
4answers
448 views

Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4} $

What is the asymptotic behavior of the sequence: \begin{equation} s_n=\sum_{k=1}^{n}k^{1/4} \end{equation} when $n\to \infty$?
7
votes
9answers
1k views

Formula for the sequence repeating twice each power of $2$

I am working on some project that needs to calculate what $a_n$ element of the set of numbers $$1, 1, 2, 2, 4, 4, 8, 8, 16, 16 \ldots$$ will be. $n$ can be quite big number so for performance issues ...
2
votes
0answers
27 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
0
votes
2answers
73 views

Test the convergence of the series $\sum_{n=1}^{\infty}a_n$

Test the convergence of the series $\sum_{n=1}^{\infty}a_n$ , where $$a_n=\begin{cases}\dfrac{1}{n^2} & \text{ if $n$ is not a square integer},\\[6pt] \dfrac{1}{n^{2/3}} & \text{ if ...
1
vote
2answers
108 views

Summing Lerch Transcendents

The Lerch transcendent is given by $$ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}. $$ While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} ...
1
vote
0answers
9 views

Flat frequency domain response

If I have a certain sequence having a flat frequency domain response, is its time domain autocorrelation a delta? In other terms, is a flat frequency response necessary and sufficient condition for ...
1
vote
2answers
58 views

Proving a theorem about Fourier coefficients

I need to prove this: Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$. Can someone please let me ...
0
votes
1answer
16 views

Coefficients of general Fourier Series

I know how to compute coefficients of Fourier Series on an interval of $2\pi$. But in this case I need to find the sine series of $f(x)=b$ on the interval $x \in [-L,L]$. Can someone please let me ...
3
votes
0answers
100 views

Using a visual “proof” to show that $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n =1$

The "proof without words" that $\sum_{n=1}^{\infty} \left(\frac 12 \right)^n =1$ is fairly well known: But why can't we apply the exact same logic to $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ ...
-4
votes
1answer
18 views

Sketching the graph of a sequence of functions [closed]

Does anyone know how to sketch graphs of a sequence of functions? Say, I need to sketch $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ for $x \in [0, 2]$. But I have no idea where to start. Can someone please help? ...
-1
votes
1answer
35 views

How to derive this inequality

I learnt that for a standard normal random variable $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
0
votes
0answers
42 views

Show that the series does not converge uniformly on $\mathbb{R}$

My Work: Here, according to the given facts $f(0)=0$ and $f$ is strictly increasing. I proved part (a) and (b) but failed to prove (c). I was going to use the definition (actually wanted to show ...
2
votes
1answer
59 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
2
votes
0answers
28 views

Order of Magnitude

It is well known (see here, for example) that we have $$ \psi\left(\frac{1}{2},T\right)=\sum_{p\leq T}\frac{1}{p}=\log(\log T)+A+O\left(\frac{1}{\log T}\right), $$ where $\psi(\sigma,T)=\sum_{p\leq ...
1
vote
3answers
185 views

Formulae for sequences

Given that for $1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}$ deduce that $(n+1)^3 + (n+2)^3 +\cdots+ (2n)^3 = \frac{n^2(3n+1)(5n+3)}{4}$ So far: the sequence $(n+1)^3 + (n+2)^3 +\cdots+ ...
0
votes
1answer
64 views

Recurrence relation: $c_{k+1}=c_k+\frac{1}{(k+1)!}$

I have no idea how to proceed solving a recurrence relation like this. I know that the terms approach $e$ but beyond that I have no idea. The relation is$$c_{k+1}=c_k+\frac{1}{(k+1)!} \ \ ; \ c_0=1$$ ...
4
votes
2answers
148 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
0
votes
2answers
75 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
1
vote
1answer
25 views

Bernoulli-like generating function

What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off. I already found the series for its ...
3
votes
2answers
106 views

The converges of $ \sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } + …=$

I would like to know wheather this series converge or diverge? $\sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } +\sqrt { 2-\sqrt { 2+\sqrt { 2+\sqrt { 2 } } } } ...
0
votes
1answer
20 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
32
votes
6answers
919 views

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} = \sum_{n=1}^{n} \frac{1}{k}$. I'm interested in knowing how to show that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}.$$ I tried ...
30
votes
3answers
851 views

Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$

I found the following formula $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$ and it is cited that Euler proved the ...
2
votes
3answers
153 views

How to prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier Series [duplicate]

Can we prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier series?
3
votes
2answers
70 views

$\int_a^b f(x)g(x)dx = \sum \int_a^b f_n(x)g(x)dx.$

Let $\sum f_n(x) $ be uniformly convergent to $f(x)$ on $[a,b]$ where each $f_n$ is continuous on $[a,b]$. If $g: [a,b] \to \mathbb R$ be integrable on $[a,b]$, then $$\int_a^b f(x)g(x)dx = \sum ...