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.

4
votes
2answers
109 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
83 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
67 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
40 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
80 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 ...
6
votes
0answers
121 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
43 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
74 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
36 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
61 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
18 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
293 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 [closed]

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
58 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
94 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
81 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
180 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
76 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
47 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 ...
0
votes
1answer
30 views

Determining composite numbers by using geometric progression. [duplicate]

I have a problem which involves finding composite numbers among given set of numbers of the form $11111...1$$(n-digits)$. Which of the following numbers is/are composite (i) $11111...1$$(91-...
3
votes
2answers
71 views

Convergence in p-adics and reals

I have been trying to solve three closely related problems about convergence of sequences in the p-adic numbers. I managed to solve two, but got stuck on the last one. Apart from trying to find a ...
0
votes
2answers
30 views

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$ I believe that I have to use the cauchy product? But how do transform the expression to be a product ...
0
votes
1answer
47 views

What is the sum of the series? $\sum_{i=0}^{K+N-3}\left\lfloor \frac{i}{N-1} \right\rfloor$

What is the sum of the series given by $$\sum_{i=0}^{K+N-3}\left\lfloor \frac{i}{N-1} \right\rfloor$$ Where $N \in [2,100000)$ and $k \in [1,100000)$ Please help me find the first term and last term ...
1
vote
1answer
63 views

Can't find the limit [duplicate]

$$\lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n}=\text{?}$$ I tried to use formulas $$\sum_{p=0}^k\binom{m+p}m = \binom{m+k+1}...
0
votes
1answer
49 views

Convergence of the sequence $\left\{x_n\right\}$ where $x_n = \frac{1}{1.3}+\frac{1}{2.5}+…\frac{1}{n(2n+1)}$ [duplicate]

Let $\left\{x_n\right\}$ be a sequence where $x_n = \frac{1}{1\cdot3}+\frac{1}{2\cdot5}+...\frac{1}{n\cdot(2n+1)}$ I have to calculate, to which point does the sequence $\left\{x_n\right\}$ converge, ...
-4
votes
1answer
84 views

Set $\int_0^1{x^n\sqrt{1-x^2}}dx\quad (n=0,1,2,…)$. Prove that the sequence $(a_n)$ is monotonically decreasing; [closed]

Set $a_{n}$=$\int_0^1{x^n\sqrt{1-x^2}}dx \quad (n=0,1,2,....)$ $ 1.$ Prove that the sequence $(a_n)$ is monotonically decreasing; $2.$ Prove $a_{n}$ = $\frac{(n-1)}{(n+2)}a_{n-2}\quad (n=2,3,…)$ ...
0
votes
1answer
23 views

How accurate is this sequence of flips in predicting the next binary number?

I saw this video on how to teach binary number on Twitter: https://mobile.twitter.com/MichaelGalanin/status/1140072321006428160 and i noticed is that the binary correspondent to the N numbers change ...
2
votes
2answers
81 views

How to compute $\sum_{k=1}^{\infty}{\frac{1}{k^2+2k}}$?

To whom this may concern, i am struggling with partial sum formulas. I don't really get why you would need to perform a partial fraction decomposition or how you know that you have to. I started by ...
0
votes
1answer
24 views

Bounded below to imply the other sequence is bounded above

Suppose I proved that $a_n=(1-\frac{1}{n})^n$ has a tail that is bounded below. Show that $b_n=(1+\frac{1}{n})^n$ has a bounded above tail. My attempt: Suppose $(1-\frac{1}{n})^n \geq m$ $\forall n &...
31
votes
4answers
629 views

Iterated integral question

Show $$\lim_{n \to\infty} \int_0^1 \cdots \int_0^1 \int_0^1 \frac{ x_1^2 + \cdots + x_n^2}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n = \frac 2 3.$$ Not sure how to start off this iterated integral ...
2
votes
1answer
18 views

Sum and ratio of convergent sequence

A sequence of real numbers $($$x_n$$)$ converges to $x$. Consider the following claims: (i) The sequence $(x_{n+1}/x_n)$ converges to $1$. (ii) The sequence $(x_{n+1} + x_n)$ converges to $2x$. ...
0
votes
1answer
25 views

convergence of series and sum and series

Say $\sum a_n+b_n$ converges then will it automatically mean $\sum a_n$ converges as well as $\sum b_n$, too? If $\sum a_n$ and $\sum b_n$ both converge will it be the case that $\sum a_n-b_n$ ...
7
votes
2answers
295 views

How to compute this constant with high precision $\sum_{n=1}^\infty \left(\frac{1}{a_n}-\frac{1}{(n+1) \ln (n+1)} \right)$

I'm interested in finding the following constant: $$b=\sum_{n=1}^\infty \left(\frac{1}{a_n}-\frac{1}{(n+1) \ln (n+1)} \right)$$ Where: $$a_1=2$$ $$a_{n+1}=a_n+\log a_n$$ This is related to my ...
2
votes
3answers
133 views

How to prove that $\sum\limits_{k=0}^{\infty}\frac{(4k-1)!!}{(2k+1)!\cdot2^{4k+1}}=\frac{\sqrt{3}-1}{\sqrt{2}}$

How to prove that $$\sum\limits_{k=0}^{\infty}\frac{(4k-1)!!}{(2k+1)!\cdot2^{4k+1}}=\frac{\sqrt{3}-1}{\sqrt{2}}$$ I need any hint to start to prove it. Thanks for any help.
0
votes
3answers
68 views

Prove monotonicity of $a_n=\left(1-\frac{1}{n}\right)^n$

$$a_n=\left(1-\frac{1}{n}\right)^n$$ I tried to compute its derivative and find their consecutive differences but didn't go anywhere.
0
votes
1answer
19 views

Cauchy-Hadamard with Triple Powers

I'm trying to determine the convergence radius of this series: $$ f(x)=\sum_{n=1}^{\infty}{(-1)^{n+1}\frac{x^{3n}}{3n\sqrt{n}\cdot8^n}} $$ I defined $$ a_n= \begin{cases} \frac{(-1)^{n+1}}{3n\sqrt{...
1
vote
3answers
65 views

Show that $F: X \to \ell^{2}$ is surjective

Let $X$ be a infinite dimensional Hilbert Space that is separable and has $\{e_{n}\}_{n \in \mathbb N}$ a countable Orthonormal Basis. I have been given the map: $F: X \to \ell^{2}, x\mapsto (\...
-1
votes
2answers
31 views

Show that the sequence is not bounded above

I must show that the sequence is not bounded above: $a_n =\frac{n^n}{n!}$, I tried to use proof by contradiction: suppose there is some $k$ such that $a_n\le k$, then $n^n \le kn!$, $n*n*n...*n \le ...