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

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1
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3answers
141 views

Computing the sum of an infinite series

I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}.$$ I tried splitting the fraction into two parts, i.e. ...
1
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2answers
38 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
0
votes
0answers
11 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
2
votes
1answer
43 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
6
votes
5answers
128 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
0
votes
1answer
110 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
1
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0answers
20 views

Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ ...
1
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1answer
44 views

Rearranging a series' terms

So i am asked to rearranje the terms in this series: $$ \sum_{i=1}^\infty \frac{(-1)^{n+1}}{n} = 1- \frac 12 +\frac13-\frac14+... $$ so that the sum of the series is equal to 0. I've seen the ...
3
votes
1answer
36 views

Does this sequence of functions converge uniformly?

So the questions says, let $a_n$ be a sequences of real numbers such that $\limsup |a_n| = 0$. Let $X = [0, 1]$ and for each $n \in \mathbb{N}$ the function $\space$ $f_n :$ $X \mapsto ...
1
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2answers
46 views

How to show that the following series converges to 1

Let $f$ be a function on $\mathbb{R}$, non-zero only on $[0,2)$. In particular $f(x)=1,x\in[0,1]$ and decreasing to zero, starting from $x=1$. Let $g(x)=f(x)-f(2x)$. Show that $$\sum_{j=0}^\infty ...
1
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2answers
54 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
0
votes
2answers
65 views

The Laurent Series of $\dfrac{e^z}{z^2-1}$

The Laurent Series of $\dfrac{e^z}{z^2-1}$ At $z=1$ As we seek for powers of $z-1$, note that: $$e^z=e\cdot e^{z-1}=e(1+(z-1)+\dfrac{(z-1)^2}{2!}+\dfrac{(z-1)^3}{3!}+...)$$ So: ...
0
votes
0answers
12 views

What are the coefficients of Modified Fourier Bessel series?

What are the co-efficients of Modified Fourier Bessel series? $$\phi g(r,z,Φ) = \sum_{v=1}^∞ \sum_{m=1}^∞ \sin\left(\frac{mπz}{L}\right) I_v \left(\frac{mπr}{L}\right)\left(A_{vm} \cos(vΦ) + B_{vm} ...
3
votes
1answer
51 views

Proving that a sequence converges or diverges [on hold]

Prove or disprove that there is a sequence $n_k$ of positive integers such that both $\cos(2n_k!)$ and $\sin(2n_k!)$ converge. I think that the series diverges but I am not sure how to prove it.
1
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1answer
34 views

$ |x_{n} - y_{n}| < \frac{1}{n} \Rightarrow |x'_{n} - y'_{n}| < \frac{1}{n}$

This came from a proof on uniform continuity theorem. My textbook claims that if sequence $(x_j)$ and $(y_j)$ on a compact set D have the condition that $ |x_n - y_n| < \frac{1}{n}$, and $(x'_j)$, ...
3
votes
0answers
115 views

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot (n+1) \cdot (n+2)}$ [duplicate]

I am trying to find the sum of the following series $$\sum_{n=1}^{\infty} \frac{1}{n \cdot (n+1) \cdot (n+2)}$$ I've already figured it out that it converges and, as it looks like a telescoping ...
2
votes
3answers
102 views

Find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$

How to find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ in the formal way? Numerically its value is $\approx 0.0217326$ and the partial sum formula contains the first derivative of ...
5
votes
0answers
103 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
0
votes
1answer
17 views

Finding the bounds for a truncation error

I have two series, $S$ and $T$ which approximate $\pi$ such that $$S_n = 4 \sum_{i=1}^n \cfrac{-1^{i+1}}{2i-1}$$ and $$T_n = \Big(12 \sum_{i=1}^n \cfrac{-1^{1+i}}{k^2} \Big) ^{\frac{1}{2}}$$ It is ...
0
votes
0answers
12 views

What does “empirical error” mean in this context?

I recently sat an exam for computational mathematics. The question asked for us to: "Write the empirical error in $\mathcal{O}(n^{-p})$ where $p$ is some integer" We were given a series $$S = 4(1 - ...
1
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
7
votes
2answers
235 views

Why does $\int_0^{\infty}\frac{\ln (1+x)}{\ln^2 (x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?

I'd like to see the reason why $$\int_{0}^{\infty}\frac{\mathrm{ln}(1+x)}{\mathrm{ln}^2(x)+\pi^2}\frac{dx}{x^2}=\gamma$$ where $\gamma$ is the Euler-Mascheroni constant. I don't have any 'neat ...
0
votes
0answers
18 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 ...
14
votes
14answers
489 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 ...
1
vote
2answers
33 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
1
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1answer
32 views

Asymptotic Estimates for a Strange Sequence

Let $a_0=1$. For each positive integer $i$, let $a_i=a_{i-1}+b_i$, where $b_i$ is the smallest element of the set $\{a_0,a_1,\ldots,a_{i-1}\}$ that is at least $i$. The sequence ...
2
votes
2answers
137 views

Rewriting sylvester's sequence in a closed form

Sylvester's sequence is defined as http://upload.wikimedia.org/math/1/6/f/16feba8ab6368dc9d965dbec35e445bb.png but according to wikipedia and wolfram mathword, this can be rewritten in closed form ...
2
votes
1answer
21 views

Absolute value of infinite series sum

How does it come about that $$\left|\Sigma_{n=-N}^{N}c_n(f)e^{inx} - \Sigma_{-\infty}^{+\infty} c_n(f)e^{inx}\right| = \left|\Sigma_{|n|>N} c_n(f)e^{inx}\right|?$$ What happens with the $n$-index? ...
1
vote
1answer
20 views

Summation of infinite series, where difference in consecutive denominator forms an A.P.

What is the sum of an infinite series where each term can be written as $\frac{p}{q}$, where p=1 always the difference between 2 consecutive denominators forms an A.P. for example $\frac{1}{2}$, ...
0
votes
2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
3
votes
4answers
1k views

Find the number of terms common in two sequences

How many terms do the two sequences $S_1$ and $S_2$ have in common? $S_1 = 1, 3, 6, 10, 15\dots$ up to $200$ terms. $S2 = 3, 6, 9, 12, 15\dots$ up to $200$ terms. I need to know the number of ...
2
votes
2answers
47 views

What's special about the cauchy product?

I am reading chapter 3 in Rudin's Principles of Mathematical Analysis. In it, Rudin defines the Cauchy product of two series. That is, given $\sum a_n$ and $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ ...
1
vote
1answer
44 views

Stadium Seating - Geometric Sequences

A circular stadium consists of sections as illustrated, with aisles in between. The diagram show the tiers of concrete steps for the final section, Section K. Seats are to be place along every step, ...
0
votes
2answers
37 views

Sum of Converging sequence

I'm given this sequence where it goes: $$ 1,\; \frac1a,\; \frac1{a(a+b)},\; \frac1{a^2(a+b)},\; \frac1{a^2(a+b)^2}, \frac1{a^3(a+b)^2}, \dotsc $$ where $a$ and $b$ are any positive integers How ...
4
votes
2answers
98 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...
18
votes
1answer
452 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
4answers
85 views

Convergence of $\sum\limits_{n=2}^\infty \ln\left(\dfrac{n^2}{n^2-1}\right)$

I made sure it passed the nth term test. Next I thought the easiest way, given that it's wrapped in ln, would be to use log rules to make it $\ln(n^2)-\ln(n^2-1)$ and then compare it to $\dfrac ...
6
votes
3answers
211 views

Infinite Series $\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}$

How can we prove the following formula? $$\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}=\frac{-2m+\sqrt{mn}\pi\cot\left(\sqrt{\frac{n}{m}}\pi\right)+n\pi^2\csc^2\left(\sqrt{\frac{n}{m}}\pi\right)}{4mn^2}$$ ...
14
votes
0answers
114 views
+50

show this sequence inequality $x_{2^n}$

Define the sequence $\{x_{n}\}$ recursively by $x_{1}=1$ and $$\begin{cases} x_{2k+1}=x_{2k}\\ x_{2k}=x_{2k-1}+x_{k} \end{cases}$$ Prove that $$x_{2^n}>2^{\frac{n^2}{4}}$$ I have ...
1
vote
2answers
27 views

series comparison test

Is this correct? Q:Determine $\sum_1^n$$\frac{2}{3+5n}$ converges or diverges. A:$\frac{2}{6n}$ < $\frac{2}{3+5n}$ , since $\sum_1^n$$\frac{2}{6n}$ is a harmonic series and diverge, then ...
1
vote
1answer
66 views

Question about two sequences with a common limit

Suppose $a _n$ is a sequence of positive integers such that $ \lim_{n \rightarrow \infty} (a_n)^{\frac{1}{n}} $ exists. Suppose there exists a sequence of positive integers $ b_n $ such that $$ a_n = ...
89
votes
8answers
9k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
2
votes
1answer
50 views

Infinite Telescoping Sum: $\sum_{i=1}^{\infty} (X_i - X_{i-1})=$?

Let $(X_i)_{i \geq 0}$ be any countable sequence of numbers and suppose that a limit exists, so $$\lim_{i \rightarrow \infty} X_i = x.$$ Consider $\sum_{i=1}^{\infty} (X_i - X_{i-1})$. Is this ...
1
vote
0answers
26 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
2
votes
2answers
57 views

Generate all De Bruijn sequences

There are several methods to generate a De Bruijn sequence. Is there a general algorithm to create all unique (rotations are counted as the same) De Bruijn sequences for a binary alphabet of length ...
0
votes
3answers
83 views

Find the formula of the sum of $\frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + \dots + \frac{1}{(1+x^2)^n}$

How would I find the sum of this geometric series: $$ \frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + \dots + \frac{1}{(1+x^2)^n} $$ I want a formula, in the form of $\frac{n}{n+1}$, that can be proven by ...
0
votes
0answers
33 views

Additional explanation needed for the solution to a spesific sequence

Some of my attempts include trying to use the the term formula for geometric sequences and some other manipulations in hope of getting a more clearer, workable expression, though without success. ...
4
votes
2answers
63 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
0
votes
1answer
35 views

Show that if $\sum_{k=1}^m c_k =0 $, $\sum_{n=0}^{\infty} \sum_{k=1}^m \frac{c_k}{nm+k} $ converges.

This is a generalization of this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my solution. To show that if $\sum_{k=1}^m c_k =0 $, ...
9
votes
2answers
880 views

Is the product of two monotone sequences monotone?

Question: The product of monotone sequences is monotone, T or F? Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing. CASE I: Suppose ...