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

learn more… | top users | synonyms (5)

14
votes
14answers
552 views
+100

New Idea to prove $1+2x+3x^2+\cdots=(1-x)^{-2}$

If $|x|<1 $ prove that $\\1+2x+3x^2+4x^3+5x^4+...=\frac{1}{(1-x)^2}$ 1st proof:suppose ...
2
votes
0answers
56 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
3
votes
1answer
123 views

Limit of quotient of two infinite series $\left(\frac{0}{0}\right)$

Let $\sum_{n=1}^\infty a_{n,k}<\infty$. I want to calculate $$L=\lim_{k\to \infty}{\sum_{n=1}^\infty a_{n,k}\over\sum_{n=1}^\infty a_{n,k+1}}$$ if I know that $\lim_{k\to \infty} ...
2
votes
1answer
45 views

Lim sup/inf of average value

Consider $$f(t)= \frac{1}{t} \int_{0}^t \sin(e^s) ds.$$ What is $$\mathrm{lim \ inf}_{t \rightarrow \infty} f(t)$$ and $$\mathrm{lim \ sup}_{t \rightarrow \infty} f(t)?$$ Using $u$-substitution, ...
5
votes
1answer
90 views

How to evaluate this double infinite sum (Catalan number)

Let $C_n = \dfrac{1}{n+1}\binom{2n}{n}$. Is it possible to find the exact value of this infinite sum ? $$\sum_{n=1}^\infty \sum_{k=n}^\infty ...
1
vote
3answers
65 views

Finding convergence of a series using integral test

The series:$$\sum_{n=1}^{\infty}\left(\frac{\ln(n)}{n}\right)^{2}$$ Question: a) show that it converges b) find the upper bound for the error in approximation $s\approx s_{n}$ Trial: The section ...
1
vote
1answer
29 views

Asymptotic Expansion of $\ f(x)=\frac{\log(x)}{\frac{\log(x)}{2\alpha}-\log(\log(x))}$

I'm looking for the asymptotic expansion as $\ x \rightarrow \infty$ for $\ f(x)$ for small $\alpha$. Ideally, I'd like to get the asymptotic expansion for all orders. How would I go about doing this? ...
0
votes
1answer
14 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 ...
2
votes
1answer
58 views

How to find the Summation S

Given function $f(x)=\frac{9x}{9x+3}$. Find S: $$ S=f\left(\frac{1}{2010}\right)+f\left(\frac{2}{2010}\right)+f\left(\frac{3}{2010}\right)+\ldots+f\left(\frac{2009}{2010}\right) $$
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 ...
11
votes
5answers
1k views

Sum of an infinite series $(1 - \frac 12) + (\frac 12 - \frac 13) + \cdots$ - not geometric series?

I'm a bit confused as to this problem: Consider the infinite series: $$\left(1 - \frac 12\right) + \left(\frac 12 - \frac 13\right) + \left(\frac 13 - \frac 14\right) \cdots$$ a) Find the sum $S_n$ ...
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) ...
2
votes
1answer
76 views

Can Eisenstein Series output complex numbers?

The 2nd Eisenstein Series, defined by, $$E_{2}(\tau)=1-24\sum_{n=0}^{\infty} \frac{nq^{2n}}{1-q^{2n}},$$ where $q=e^{i\pi \tau}$ is the nome acts on upper-half plane. Must it always output real ...
1
vote
0answers
69 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 ...
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
17 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 ...
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 ...
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 ...
-1
votes
0answers
51 views

Nice formula for $N*(N-1)*(N-2)*…*(N-(N-1))$? [closed]

Suppose $N\in\mathbb{N}$. Does the formula $S = N*(N-1)*(N-2)*...(N-(N-1))$ have some nice and compact form?
1
vote
2answers
50 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 ...
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?
2
votes
2answers
34 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
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
0answers
33 views

how to solve for a term in the denominator of a summation? [closed]

For example, if I have: $$\displaystyle \sum_{i=1}^n \frac{1}{t_i + x} = \cdots$$ is there a way I can extract $x$ out of the summation?
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
57 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 ...
2
votes
4answers
382 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.
0
votes
0answers
21 views

Sum of series involving factorials [closed]

$$ \sum\limits_{j=0}^{[\frac{n}{l}]}(-1)^{slj}\left( \begin{array}{c} n \\ lj \\ \end{array} \right)^s \frac{x^{kj}}{[(a)_{bj}]^s}, $$ where $l,s,n,k,a,b$ are natural numbers and x is ...
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 ...
0
votes
1answer
17 views

Does the series $\displaystyle\sum_{j=1}^\infty -\log (1-p_j^{-3/4})$ diverge, where $\{p_j\}$ is the set of primes in increasing order?

Here, $\log$ is the natual logarithm. Is there a simple convergence test I can use? Thanks.
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 $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ ...
-3
votes
1answer
16 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? ...
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
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 ...
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 ...
4
votes
2answers
125 views

Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
1
vote
2answers
96 views

Find a closed form of the series $\sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2}$

The question I've been given is this: Using both sides of this equation: $$ \frac{x}{1-x} = \sum_{n=1}^{\infty}x^n $$ Find an expression for: $$ \sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2} $$ Any help ...
1
vote
3answers
184 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
2answers
74 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 ...
0
votes
1answer
41 views

Geometric sequence, find the common ratio [closed]

Given: Sum of first five terms is $44$, sum of the next five terms is $-11/8$. Find common ratio and the first term of the series. Also, find the sum of infinity.
0
votes
1answer
19 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 ...
0
votes
2answers
37 views

convergence of the series for $p>0$ , $\sum n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$

Discuss the convergence of the series for $p>0$ , $$\sum n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$$ I tried through ratio test. But it fails. I think it will be by comparison ...
0
votes
1answer
19 views

Combinatoric for number of ways to have monotone-increasing sequence

I hope I am using the right term. By monotone-increasing I mean to imply that it is a non-decreasing sequence. So for example a sequence $1, 1, 2, 5, 6, 10, 10, 11$, etc. Anyhow, consider a ...
1
vote
1answer
25 views

Applying squeeze theorem to conditionally convergent series

Suppose that the series ∑_n≥1(a_n) converges conditionally. Then by the Riemann Series theorem, for any real number L there exists a rearrangement of a_n(let's call it b_n) that converges to L. For a ...
2
votes
1answer
227 views

Partial sum of exponential series strictly increases after certain step

While trying to show that partial exponential series evaluated at two different values are strictly increasing provided that sufficient number of terms are applied I stuck at a problem. Given two ...
0
votes
1answer
53 views

Integer series will become constant? [closed]

Given a bounded integer sequence $(a_n)_{n\in\mathbb{N}}$, prove or disprove that if the sequence $(a_n + n)_{n\in\mathbb{N}}$ has no repetitive elements: $$ \forall n,m \in \mathbb{N}, \,\,m \neq n ...
0
votes
1answer
61 views

Convergence of the series $\sum_{n=0}^\infty \sin(n! \pi m \sin(1))$

In this exercise I was asked to prove the convergence of the following infinite sum: $$\sum_{n=0}^\infty \sin(n! \pi m \sin(1)),$$ where $m$ denotes any integer. I don't have any idea on how to ...
1
vote
0answers
28 views

Is this :$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $ irrational series for every natural number $k$?

Is this: $$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $$ irrational series for every natural number $k$? Where : $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of ...
1
vote
1answer
30 views

General methods of solving non-linear recurrences

Let's consider a sequence $$ \{a_{n} \} _{n=0}^{\infty}, a_{0}=1, a_{n+1}=\frac{a_{n}}{1+na_{n}}$$. Taking $$b_{n}=\frac{1}{a_{n}},$$ we bring the exact reccurence to the following one: $$b_{0}=1, ...