Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

0
votes
2answers
15 views

Need an explanation on a certain terminology about converegence range in series.

I found x's possible values, but I am also required to check the edges. What does it mean to check the edges? and why is it necessary for solving the problem?
0
votes
3answers
62 views

Proving that $a_n$ is not monotonically increasing if $a_n\ge 0$ and $\lim a_n=0$

Had an exam today and there was the following question: Let $a_n$ be a sequence such that for all $n\in \mathbb N$ we have $a_n \geq 0$ and $\lim a_n = 0$. Prove that $a_n$ is not monotonically ...
2
votes
2answers
791 views

Progressive Roulette Betting

In this problem we discuss a betting strategy known as "progressive betting". Here's the setup: Bets are repeatedly made at a roulette according to the following strategy: All bets are made the same ...
0
votes
1answer
26 views

how to interchange sum of series formulas?

I have this formula \begin{equation} \sum_{k=a+1}^{b}\frac{b-k+1}{b-a+1} \end{equation} I want to convert it to \begin{equation} \frac{1}{b-a+1}\sum_{k=1}^{b-a}k \end{equation} What are the ...
-1
votes
2answers
97 views

A recurrence relation for Fibonacci squared

The sequence $S_n$ is defined as $$S_1=S_2 =1$$ and for $n\ge 2$, $$S_{n+1}=2(A_n + G_n)$$ where $A_n = \frac {S_n+S_{n-1}}{2}$ is the arithmetic mean and $G_n= \sqrt { S_nS_{n-1} }$ is the geometric ...
0
votes
1answer
20 views

Proving that a sequence in $\ell^2$ is a Cauchy sequence

Let $(x^{(n)})_{n\in\mathbb{N}}, x^{(n)}:= \sum\limits_{i=1}^n \frac{1}{i} e_i$, where $e_i$ is the sequence that is $0$ everywhere but $1$ in the $i^{th}$ element. I would like to prove that this ...
0
votes
1answer
43 views

If $a_n$ monotonic, $a_n>0$, and $\sum\sqrt {a_na_{n+1}}$ converges, prove $\sum a_n$ converges. [duplicate]

need help solving this problem: Given $a_n$ monotonic, $a_n>0$. prove that if $\sum\sqrt {a_na_{n+1}}$ converges then $\sum a_n$ converges.
0
votes
3answers
39 views

Proving that $\sum_{n=1}^\infty\left(\sqrt[n]{e}-1-\frac 1n\right)$ converges

Problem: Prove that the series $$S=\sum_{n=1}^\infty\left(\sqrt[n]{e}-1-\frac 1n\right)$$ converges. My idea is to change $\sqrt[n]{e}$ to $\sqrt[n]{\left(1+\dfrac 1n\right)^n}$, and then try to ...
3
votes
2answers
2k views

Is $\sum_{n=3}^\infty\frac{1}{n\log n}$ absolutely convergent, conditionally convergent or divergent?

Classify $$\sum_{n=3}^\infty \frac{1}{n\log(n)}$$ as absolutely convergent, conditionally convergent or divergent. Is it, $$\sum_{n=3}^\infty \frac{1}n$$ is a divergent $p$-series as $p=1$, and $$\...
1
vote
2answers
71 views

About series $\sum_{n=1}^\infty\frac{a_n}{n^s}$

Can you see if (a), (b) and (c) ar correct, and give a hint for (d). Thabk you very much. Let $(a_n)$ be a bounded sequence of real numbers. (a) Show that the series $\sum\limits_{n=1}^{\infty}\...
4
votes
2answers
234 views

Is it true that $\sum_{n = 0}^\infty \frac{a_n}{n}$ is also convergent? [duplicate]

Let $\{a_n\}_{n = 0}^\infty$ be a sequence of real numbers such that the series $\sum_\limits{n = 0}^\infty |a_n|^2$ is convergent. Is it true that $\sum_\limits{n = 0}^\infty \dfrac{a_n}{n}$ is also ...
1
vote
1answer
84 views

How to solve this equation for $y$?

I have the following equation that I would like to solve for $y$: $$\alpha=\sum_{i=1}^{n}\exp(-y\beta_i)$$ $\alpha$, $y$ and $\beta_i$ are all positive reals (the $\beta_i$'s are independent random ...
1
vote
2answers
56 views

Show that sequence converges of $r_2$

We have the function $f(x)=x^2-x-12=0$ with roots $r_1=-3$ and $r_2=4$. We consider the sequence $x_{n+1}=g(x_n), \ n=0,1,2,\ldots $ where $g(x)=\sqrt{x+12}$. We want to show that $x_n\rightarrow ...
-1
votes
2answers
88 views

Convergence of the real series $\sum_{m=1}^\infty \sum_{n=1}^\infty\frac{1}{(m+n)^2}$. [on hold]

Consider the two array series $$\sum_{m=1}^\infty \sum_{n=1}^\infty\frac{1}{(m+n)^2},$$ does it converge? Can we find some estimate for the general term $\dfrac{1}{(m+n)^2}$ so that we can apply ...
-2
votes
0answers
31 views

Convergent and divergent series of real numbers. [on hold]

$$\sum_{m=1}^ \infty\sum_{n=1}^\infty\frac{1}{(m+n)^2}$$ How to prove the above series is convergent or divergent by easiest and simple way.
6
votes
0answers
61 views

Verifying this limit of a sum

A lenghty limit, $$\lim_{ n \to \infty}-\sqrt{8}\cdot \frac{n!}{B_n}\sum_{j=0}^{n}\frac{(1-2^{1-j})(1-2^{1+j-n})B_{n-j}B_j}{4^j(n-j)!j!}=\pi$$ The limit seems to appraoches to $\pi$ but I am not ...
0
votes
0answers
24 views
+50

$\lbrace (\alpha,\beta)\in \mathbb{R}^2: Q \mbox{ is the } Q\mbox{-matrix of a non-explosive continuous time Markov chain}\rbrace$

Let $\alpha,\beta\in \mathbb{R}$, $E=\lbrace 0,1,2,...\rbrace$ and $Q=(q(x,y):x,y\in E)$ given by \begin{align} q(0,n)=\begin{cases} -1 &\mbox{ if } n=0 \\ 1 &\mbox{ if } n=1 \\ 0 &\mbox{ ...
-3
votes
0answers
30 views

Find the value of $(a^2+b^2+c^2)(b^2+c^2+d^2)$? [on hold]

If $a, b, c, d$ are in G.P., then what is the value of $(a^2+b^2+c^2)(b^2+c^2+d^2)$?
-1
votes
2answers
36 views

How to find arithmetic mean of number [on hold]

The arithmetic mean of 1,8,27,64... upto n terms is given by .... I know the formula for arithmetic mean But i don't know how to apply it I applied a+b/2 But it's not logical here so Plzz tell me ...
4
votes
2answers
105 views

Prove $\forall m\in\mathbb{N},m\neq1:\quad\sum_{n=1}^{m}\frac{1}{n^2}\leq\int_1^m\frac{\sqrt{x^6+4}}{x^3}\ dx$

So I have stumbled upon this question and was very intrigued on how to solve it. I have an intuitive solution, but I guess that's not enough. I would be glad if you could shed some light on how to ...
5
votes
4answers
62 views

Finding floor of reciprocal sum

Evaluation of $$\bigg \lfloor \frac{1}{\sqrt[3]{1}}+\frac{1}{\sqrt[3]{2^2}}+\frac{1}{\sqrt[3]{3^2}}+\cdots +\frac{1}{\sqrt[3]{(1000)^2}}\bigg\rfloor$$ Where $\lfloor x\rfloor $ is the floor ...
0
votes
1answer
80 views

Limit of infinite product: (3n+1)/(3n+2)

I am struggling to find the following limit: $$\lim_{n\to \infty} \frac{1\cdot4\cdot7\cdot...\cdot(3n+1)}{2\cdot5\cdot8\cdot...\cdot(3n+2)}$$ So far I tried to use squeeze theorem and logarithm to ...
1
vote
2answers
64 views

Equality of 2 forms of summation - derangement problem

The number of ways of placing $n$ objects not in position is given by the inclusion-exclusion number $D_n$: $n! \left( 1-\dfrac{1}{1!}+\dfrac{1}{2!}+....+(-1)^n\dfrac{1}{n!} \right)$ which can also ...
1
vote
0answers
37 views

Digits of irrational numbers, organized chaos, modular arithmetic, and uniform equilibrium distribution

Is there a simple increasing sequence $\{ x_n \}$ of positive integers such that $x_n \mbox{ mod } b$ is the $n$-th digit of some known mathematical irrational constant in some integer base $b$? ...
0
votes
0answers
18 views

Geometry / Calculus - relationship between surface area and volume of a burrito

Basically, trying to find the relationship between the radius of the surface and the volume of a burrito as a (graph-able?) mathematical progression in terms of radius (or width) of the tortilla. For ...
4
votes
0answers
79 views

How to solve the recurrence relation $a_{1}=2, a_{n}=\frac{a_{n-1}+2}{2 a_{n-1}+1}(n \geq 2)$ with generating functions?

There's already a way to solve it, called "fixed point method", that is, from the relation we define its characteristic equation as $x=\dfrac{x+2}{2x+1}$,then we have $x_1=1,x_2=-1$. So the following ...
2
votes
3answers
74 views

Prove that $\lim_{j \to\infty}\sum_{k = 1}^{\infty} \frac{a_k}{j+k} = 0$

Exercise: Suppose that $a_k \geq 0$ for $k$ large and that $\sum_{k = 1}^{\infty} \frac{a_k}{k}$ converges. Prove that $$\lim_{j \to \infty}\sum_{k = 1}^{\infty} \frac{a_k}{j+k} = 0$$ Attempt ...
5
votes
0answers
111 views

An integration-via-summation formula

For symbolic transformation of integrals and series I occasionally use this formula: $$\int_0^1f(x)\,dx=-\sum_{n=1}^\infty\sum_{m=1}^{2^n-1}\frac{(-1)^m}{2^n}f\left(\frac m{2^n}\right)\tag{$\diamond$}$...
1
vote
1answer
42 views

Arithmetic sequences. Writing the general term differently

So I'm dealing with this problem that has to do with arithmetic sequences and I was wondering if I could write the general term of this kind of sequences (finite) like $$a_{k-1}=a_1+(k-\lfloor \frac{n}...
5
votes
2answers
73 views

How can I prove this sequence converges to 1?

Suppose $0<a<1$ and define $a_n=(1+a^n)^n$, show that $a_n \to 1$ by using binomial expansion on each $a_n$ and compare to a geometric sum. I know how to compute binomial on $a_n$ but I am ...
0
votes
3answers
35 views

Combinatorics proof involving finite series

I am trying to prove the following identity with little success!: $\sum_{k=0}^{p-1} (p-k) = p(p+1)/2$. Any suggestions? Thanks.
0
votes
1answer
39 views

What strategies can I adopt for proving monotonicity of a sequence. (Do not have a problem in mind but I am open to discussions)

So I have been thinking about different ways/strategies of proving monotonicity of sequences, as I said in the title, I do not have a problem in mind but I want to hear different ways I may use during ...
12
votes
3answers
3k views

How to generalize the Thue-Morse sequence to more than two symbols?

The Thue-Morse sequence is defined as a binary sequence and can be generated like 0, 01, 01 10, 01 10 10 01, 01 10 10 01 10 01 01 10, ... . So the second ...
0
votes
1answer
60 views

Express $a_n$ with: $a_1, a_2, a_3, n.$

$a_{n+3} - 3a_{n+2} + 3a_{n+1} - a_n = 1.$ Express $a_n$ with: $a_1, a_2, a_3, n.$ Hint: Create a new series that's defined like this: $x_n = a_{n+2} - a_{n+1}.$ What I have discovered so far: $$x_n ...
4
votes
1answer
72 views

There exists a continuous function $f$ such that $f(\Bbb Q) \subseteq \Bbb R\setminus\Bbb Q$ and $f(\Bbb R\setminus\Bbb Q)\subseteq\Bbb Q$ [duplicate]

True or false: There exists a continuous function $f: \Bbb R \to \Bbb R$ such that $f(\Bbb Q) \subseteq {\Bbb R}\setminus {\Bbb Q}$ and $f({\Bbb R}\setminus {\Bbb Q}) \subseteq {\Bbb Q}$. My attempt:...
0
votes
2answers
44 views

Finding the sum for the following series

Evaluate $$ \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)3^n } $$ I can show very easily that this series converges using the alternating series test. By setting $$b_n = \frac{1}{(2n+1)3^n} \ \ \ \ \ \...
0
votes
1answer
16 views

$ Z = \sum_{1 \le r < s \le 2n}^{2n} f(r,s) x_{r} x_{s} $, How to prove the the maximum occurs at the end points?

We have the summation $$ Z = \sum_{1 \le r < s \le 2n}^{2n} f(r,s) x_{r} x_{s} $$ The possibilities of $x_{i}$s are bounded by $-1 \le x_{i} \le 1$. Also, $f(r,s)$ real function. How to prove the ...
0
votes
0answers
18 views

Is this Parentheses-Inserting, Series-Rearranging and Re-indexing argument correct?

Let $X$ be an infinite subset of $\mathbb{R}$, and let $\left\{ f_{n}\left(x\right)\right\} _{n\geq0}$ be a sequence of functions $f_{n}:X\rightarrow\mathbb{C}$ so that, for each $x\in X$, there is a ...
6
votes
2answers
289 views

Is this series for Pi correct? And who has done it before?

The idea was to use an infinite series of triangles. The red then green then the... to get the area of this sector then the area of the circle is 16 times this. If it is a unit circle than area should ...
1
vote
1answer
33 views

An investigation of Heine´s characterization of continuity

This characterization of continuity, for example, for functions $f: \mathbb R \to \mathbb R$, can be stated as: A function $f: \mathbb R \to \mathbb R$ is continuous at the point $x_0$ if and only ...
-2
votes
2answers
26 views

Two proofs concerning sequence [on hold]

Let $S$ be a non-empty subset of $\mathbb{R}$ having a limit point $l$. Show that there exists a sequence $\left\{x_n\right\}$ of distinct elements of $S$ such that $\lim x_n= l$. Let S be an ...
1
vote
0answers
57 views

Find number of sets for which $S_1 \subseteq S_2 \supseteq S_3 \subseteq … $

Find number of $$ \langle S_1, S_2, ... , S_k \rangle $$ sequences of sets (each set $S_i \subset [n]$ for which a) $S_1 \subseteq S_2 \subseteq S_3 \subseteq ... $ b) $S_1 \subseteq S_2 \supseteq ...
0
votes
2answers
18 views

How can I show that this set is a closed subspace

Let $E:= \{(x_n) \in l^{2}(\mathbb N ) \mid x_{2k}=x_{2k+1}\}$ How can I show that $E$ is a closed subvector space of $l^{2}(\mathbb N )$ ? I tried to write $E$ as the kernel of a continuous linear ...
0
votes
4answers
93 views

How to prove $a^n − b^n = (a − b) \sum_{i=1}^{n}a^{n-i} b^{i-1}\le (a − b)na^{n−1}$.

Please tell if the problem can be solved using telescoping technique or not. If yes, how to prove $a^n − b^n = (a − b) \sum_{i=1}^{n}a^{n-i} b^{i-1}\le (a − b)na^{n−1}$ using that. It is given that $...
4
votes
1answer
74 views

How to find the growth rates of $n$ bacteria, knowing the sizes of bacteria from $m$ observations?

Each bacterium grows at a some constant rate, i.e. every minute the size of the bacteria increases by some constant value. Different bacteria can grow at different rate (they can also grow at same ...
5
votes
1answer
177 views

Sum with floor of harmonic series [closed]

How can I prove the following? $$\sum_{i=1}^{n} \left\lfloor \frac{n}{i} \right\rfloor \sim n \sum_{i=1}^{n}\frac{1}{i}$$
1
vote
2answers
73 views

Evaluate $\frac {1+\frac {2^2}{2!} +\frac {2^4}{3!}+\frac {2^6}{4!} +\dots}{1+\frac {1}{2!}+\frac {2}{3!}+\frac {2^2}{4!}+\dots}$

Evaluate the given series $$\dfrac {1+\dfrac {2^2}{2!} +\dfrac {2^4}{3!}+\dfrac {2^6}{4!} +....}{1+\dfrac {1}{2!}+\dfrac {2}{3!}+\dfrac {2^2}{4!}+....}$$ If we factor out $\dfrac {1}{2^2}$ from the ...
2
votes
2answers
56 views

extending a continuous map to quotient spaces.

Let $A\subset B$ and $C\subset D$ metric spaces. Suppose that there is $f: B-A\rightarrow D-C$ a homeomorphism such that for any $x\in A$, and any converging sequence $x_{n}\rightarrow x$ such that $...
0
votes
1answer
46 views

Find the coefficient of $x^r$ in the expansion of $e^{e^x}$. [duplicate]

Find the coefficient of $x^r$ in the expansion of $e^{e^x}$. My Attempt: $$e^{e^x}= 1+\dfrac {e^x}{1!} + \dfrac {(e^x)^{2}}{2!} + \dfrac {(e^x)^{3}}{3!}+......$$ How to proceed further?
0
votes
1answer
754 views

How to approach this proof on convergence of infinite product?

I have the following question I was given on a tutorial sheet to revise Infinite Series, and on it Infinite Products are introduced, as is the following question: Assume $\,b_{n}>0\,$ for all ...