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

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55 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 ...
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1answer
31 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 $(x'_j)$, ...
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3answers
86 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 ...
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1answer
14 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 ...
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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 - ...
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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 ...
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72 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( ...
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2answers
29 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 ...
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1answer
48 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 ...
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1answer
18 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 $\frac{1}{2}$, ...
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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 $f_n$ ...
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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 $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ ...
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2answers
34 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 ...
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1answer
39 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, ...
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1answer
31 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
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1answer
20 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? ...
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2answers
26 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 ...
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2answers
94 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 ...
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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 ...
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0answers
99 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 ...
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1answer
64 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 = ...
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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. ...
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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 ...
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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 ...
2
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2answers
47 views

Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$

The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs. We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + ...
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0answers
36 views

Why is it so hard to find a generating function for Somos' sequence?

The sequence is $\{1,2,12,576,1658880,\dots\}$. The $n$th number is obtained by squaring the $(n-1)$-th number and multiplying by $n$. So we start with $a_1=1$, $a_2=1^22=2$, $a_3=(1^22)^23=12$. In ...
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0answers
17 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 ...
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3answers
39 views

Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ ...
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3answers
43 views

Does the following series converge uniformly?

I know how to show that the following series will converge absolutely. But am unsure how to show it will or will not converge uniformly for $z\in (0,1).$ $\displaystyle \sum_{n \mathop = 1}^{\infty} ...
2
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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 ...
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0answers
49 views

Counting the sum $\sum^{\infty}_{k=0} q^{k^{2}}$

Is it possible to obtain explicit form of the sum $\sum^{\infty}_{k=0} q^{k^{2}}$ (without using elliptic functions)? It is well known that $\sum^{\infty}_{k=0} q^{k} = \frac{1}{1-q}$ for all $q \in ...
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0answers
28 views

Why does this hold: $ \sum^{\infty}_{n=0}\frac{x^n}{n!}\mathbf e_{n}(y)+\sum^{\infty}_{n=1}\frac{y^n}{n!}\mathbf e_{n-1}(x) = e^{x+y} $

In the book "Stochastic Processes for Insurance and Finance" by Rolski et al. the following identity is used: $$ \sum^{\infty}_{n=0}\frac{x^n}{n!}\mathbf ...
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2answers
14 views

Series of positive factors of a number divided by that number

Let $S_n$ be the sum of the positive factors of $2015^n$, with $n$ being a positive integer approaching infinity. What is $\dfrac{S_n}{2015^n}$? I might be on the wrong track, but I figure that if $x ...
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0answers
30 views

To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
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2answers
43 views

Asymptotic Expansion of $\ f(x)=(1-\beta \frac{ log(log(x))}{log(x)})^{\beta}$

So I got this function and I'm looking for an asymptotic expansion for different values of$\ \beta > 1 $ $\ f(x)=\left(1- \beta \frac{\log \left( \log(x) \right)}{\log(x)} \right)^{\beta}$ as $\ x ...
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2answers
36 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
2
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1answer
23 views

If $\sum_{k=0}^{r-1} c_k =0 $, and $a_n \to 0$, does $\sum_{n=0}^{\infty} \sum_{k=0}^{r-1} c_ka_{nr+k} $ converge?

This is a generalization of the alternating series convergence result and this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my question: If ...
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1answer
82 views

Mistakes in $\lim_{a\to \infty}(a^2 - a) = - \frac{1}{6}$?

One can say, using Ramanujan summation or the zeta function regularization, that the sum $\sum_{k=1}^{\infty} k=- \frac{1}{12}$. Using this result I've gotten a very confusing and counterintuitive ...
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2answers
44 views

The sum of three natural numbers are $111$, and the three numbers are in geometric progression. [on hold]

Find all triples of natural numbers $(a,b,c)$ such that $a,b$ and $c$ are in geometric progression, and $a+b+c=111$. Any pointers?
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1answer
23 views

Identity for the product of sequences

Very easy question, How to express $\prod_{i=0}^n \prod_{j=0}^i a_i a_j $ as a function of just one index? Incidentally, where to find identities for product of sequences? There's a lot on ...
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1answer
121 views

Cauchy Sequences--is the floor function of a Cauchy sequence also a Cauchy sequence?

Okay so say you have some Cauchy sequence (a_n). And c_n=[[a_n]], where [[x]] refers to the greatest integer less than or equal to x. Is c_n also a Cauchy sequence? This is what I've got so far, ...
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2answers
61 views

Find the limit of this sequence

$y_0=k$ where $k$ is a constant. $x_{n+1}=30-\dfrac{y_n}{2}$ $y_{n+1}=30-\dfrac{x_{n+1}}{2}$ Prove that $(x_n, y_n)$ converges to $(20, 20)$ for all values of $k$. My attempt: I wrote a computer ...
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1answer
26 views

Prove that $T_n(x)={}_2F_1\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) $

Prove that, for Chebyshev polynomials of the first kind, \begin{align} T_n(x) & = \tfrac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} ...
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0answers
19 views

Can convolution be used to measure the difference between two sequences?

Say I have an infinite sequence $S_1$ and another finite sequence $S_2$. If I calculate $$ E = S_1 ∗ S_2 $$ does it somehow reflect whether $S_2$ appears somewhere in $S_1$? What if an approximate ...
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105 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 ...
2
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1answer
45 views

Proving the Fibonacci sum $\sum_{n=1}^{\infty}\left(\frac{F_{n+2}}{F_{n+1}}-\frac{F_{n+3}}{F_{n+2}}\right) = \frac{1}{\phi^2}$ and its friends

In this article, (eq.92) has, $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+1}F_{n+2}} = \frac{1}{\phi^2}\tag1$$ and I wondered if this could be generalized to the tribonacci numbers. It seems it can ...
0
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1answer
34 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 $, ...
3
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0answers
42 views

Reorder this series to change its sum [duplicate]

If in the series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\cdots$ the order of the terms be altered, so that the ratio of the number of positive terms to the number of negative terms in the first $n$ ...
2
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2answers
22 views

Is the series $\sum_{n=1}^\infty n^r \exp(-k\sum_1^n \frac{1}{m})$ convergent when $k>r$ and $r<k$?

The series is: $$\sum_{n=1}^\infty n^r \exp(-k\sum_1^n \frac{1}{m})$$ The problem is asked to investigate this series when $r>k$ and $r<k$. However: $$ \frac{u_{n+1}}{u_n}\approx ...
4
votes
3answers
61 views

Determine the value of $ p $ for which the following infinite series converges and for which it diverges.

Determine the value of $ p $ for which the following infinite series converges and for which it diverges: $$ \sum_{n = 2}^{\infty} \frac{\sqrt{n + 2} - \sqrt{n - 2}}{n^{p}}. $$ I don’t know how to ...