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

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4
votes
1answer
31 views

Find $\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right)$

Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$ I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro ...
0
votes
1answer
30 views

Sequence problem with common difference

I have a problem with one sequence word problem. You add $1000$ to your bank account and withdraw $62$ the first year, and withdraw $4$ more every year than the year before. I have to find how much ...
0
votes
0answers
12 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
1
vote
1answer
16 views

Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
3
votes
2answers
54 views

Limit of a sequence of averages (three variables)

Let $a_0 = 0$, $a_1 = 0$, $a_2=1$ and for $n>2$, $a_n = \dfrac{a_{n-1}+a_{n-2}+a_{n-3}}{3}$. Consider $\lim\limits_{n \to +\infty} a_n$. Using a python script I found that $a_n$ tends to ...
-1
votes
1answer
36 views

looking at the alphabet ,the letters are numbered 1-26 ,

looking at the alphabet ,the letters are numbered 1-26 , such that 1 =one=15+14+5=34 (O=15, N=14, E =5 ) 2=two=20+23+15=58 (T=20, W=23, 0=15) 3=three =56 4=four=60 ...
0
votes
1answer
31 views

To test the convergence of series 1

To test the convergence of the following series: $\displaystyle \frac{2}{3\cdot4}+\frac{2\cdot4}{3\cdot5\cdot6}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot8}+...\infty $ $\displaystyle 1+ ...
7
votes
4answers
215 views

Test the convergence of a series

To test the convergence of a series: $$ \sum\left[\sqrt[3]{n^3+1}-n\right] $$ My attempt: Take out $n$ in common: $\displaystyle\sum\left[n\left(\sqrt[3]{1+\frac{1}{n^3}}-1\right)\right]$. So this ...
0
votes
1answer
29 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
15
votes
2answers
221 views

Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.
2
votes
0answers
68 views

How prove this sequence $u_{m}=v_{m}$

Question: Assume that $m$ is a positive integer, define the sequence $$\{u_{k}\},\{v_{k}\},u_{0}=v_{0}=u_{1}=v_{1}=1$$ and for any real number $a_{i},i=\{1,2,\cdots,m-1\}$, $$\begin{cases} ...
4
votes
2answers
52 views

$\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ $\Rightarrow \sum_{n=1}^{\infty} c_n$ converges?

If $\lim_{N\to \infty} \sum_{n=N}^{2N} c_n = 0$ do we have that $\sum_{n=1}^{\infty} c_n$ converges? At first this did not seem true($\sum_{n=N}^{2N} (-1)^n$ is $0$ when N is odd), but I've failed to ...
4
votes
6answers
606 views

Sum of an unorthodox infinite series

$ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $ This is a pretty unorthodox problem, and I'm not quite sure how to simplify it. Could I get a solution? Thanks.
1
vote
1answer
42 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
3
votes
3answers
47 views

Convergence of $\sum \frac{1}{\ln n}(\sqrt{n+1} - \sqrt{n})$

Show that $$ \sum\frac{1}{\ln n}(\sqrt{n+1} - \sqrt{n})$$ Converges. I've tried the telescopic property or even write it as $$\sum \frac{1}{\ln n (\sqrt{n+1}+\sqrt{n})}$$ But didnt help. Thanks ...
5
votes
0answers
54 views

Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
1
vote
5answers
107 views

Sequence $\sin^2n$

I would like to prove that if $a_n = \sin^2(n),$ then it does not converge. Usually we show 2 subsequences with different limits for those sine cases, but I could not do this since my n is a natural ...
1
vote
3answers
41 views

How fast does a sequence with finite sum go to zero?

Suppose $z_n$ is a nonnegative sequence, monotonically decreasing to zero, and $$\sum_{i=1}^{\infty} z_i < 1.$$ Is it possible to translate this into a bound on the how small $z_i$ is? For example, ...
0
votes
1answer
20 views

how many variables are there from 9 digits excluding repeat numbers

I have the numbers 1 to 9 I need to know how many different 9 digit code variations i would have using 1-9 but excluding any "next digit" replications. example: 123456789 is acceptable 112345678 ...
0
votes
0answers
19 views

How to choose the parameter?

Can I choose Parameters $\beta \in (1,2)$ and $1 \ll x \ll n$, such that $$\sum_{i=1}^{x} i i^{-\beta} \gg \sum_{i=x+1}^{n} i i^{-\beta}$$ Would be great if you could give an example.. $\gg$ means ...
1
vote
1answer
27 views

Programming PARI/GP to do a sum

I'm trying to compute the following sum in PARI/GP $C=\sum_{n=1}^{\infty} \frac{g(n)}{n^2}$ where $g(n)$ defined as $$g(n)=(-1)^r, \qquad r=\text{number of even indexed prime factors of $n$}$$ By ...
1
vote
1answer
52 views

Can I prove this, or hopeless? Deviating too much from mean

Can I prove this: We have a sequence of vectors $\left(X_i(n)\right)$ for $i=1,\ldots,t$, where $n\rightarrow \infty$. $t$ does depend on $n$ and is Chosen such that $1 \ll t \ll n$, for instance, ...
-1
votes
0answers
19 views

Single formula sequence partitionining interval

So I have the real sequence for fixed $x\in \mathbb{R}$: $y_{j}(x)=\begin{cases}f_{j-1}(x) &\text{if } |f_{j-1}(x)|\leq |x_{j}|, \\ f_{j}(x) & \text{else}, \end{cases}$ where 1) ...
0
votes
1answer
21 views

A function relating $k$ and $j$, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^{l}2^{n-i}\leq j$ and $n= \lfloor \log_{2}j \rfloor$

Do you know any function that relates k and j, where $k=\max_{l\in \mathbb{N}}\sum_{i=0}^l 2^{n-i}\leq j$ and $n=\lfloor \log_2 j \rfloor$? So, say, for $j=3$: $n=1$ and $k=1$ because $3\geq ...
1
vote
2answers
46 views

Sequences and series

If $p, q, r$ are in G.P. and the equations: $$px^2 + 2qx + r = 0$$ $$dx^2 + 2ex + f = 0$$ Have a common root, then show that $$\frac{d}{p}, \frac{e}{q}, \frac{f}r$$ are in A.P. Well I tried taking ...
2
votes
1answer
25 views

Existence of a sequence of integers $\lbrace a_k\rbrace_{k\geq 1}$ so that the first $k$ digits of $a_k\alpha$ are $0$ where $\alpha$ is irrational.

Let $\alpha$ be an irrational number. Is there a sequence $\lbrace a_k\rbrace_{k\geq 1}$ of integers so that the first $k$ digits of the fractional part of $a_k·\alpha$ are $0$? (in base $2$, for ...
3
votes
3answers
60 views

For an irrational number $a$ the fractional part of $na$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
1
vote
1answer
48 views

4 crystal balls and a 10,000 story building

There is an analog of this question I've heard with 2 crystal balls but a higher number like 4 or more makes it much more interesting. You are given 4 crystal balls and there is a 10,000 story ...
12
votes
1answer
138 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
0
votes
0answers
30 views

Series representation for $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 \pi ^2 A^2+W^2}+6 \pi ^2 A^2+3 W^2}}{\sqrt{2}}$

My question is, is there a series representation or other function of $L$ and $A$ I can use when I solve the following equation for $W$? $L=\frac{3}{2} \sqrt{4 \pi ^2 A^2+W^2}-\frac{\sqrt{5 W \sqrt{4 ...
1
vote
1answer
70 views

interval of convergence of $e^x$

Can somebody explain how to find the interval of convergence for $$ e^x=\sum_{n=1}^\infty\frac{x^n}{n!} $$ I don't fully understand the ratio test/root test/integral test etc. and I don't understand ...
1
vote
2answers
36 views

Checking for convergence of series

To check the convergence of the series $\displaystyle \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+..\infty$ Attempt 1: Term $\displaystyle u_n= \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ ...
0
votes
4answers
37 views

If (m+n)th term of an Arithmetic Progression is p, and (m-n)th term is q, then find the mth and nth terms. [on hold]

If (m+n)th term of an Arithmetic Progression is p, and (m-n)th term is q, then find the mth and nth terms.
1
vote
2answers
27 views

Exact solution for rate parameter in finite geometric series

All, I'm working on a problem whereby, given a known quantity, Q, and number of intervals, n, I want to calculate the rate parameter -1 < r < 1 of a finite geometric series. Is there a way to ...
3
votes
2answers
72 views

Closed form of a complex series sum

I am working on a proof that require a closed form (if that is not possible then at least a tight lower bound) of the expression below: $$A(n,k)=\sum_{i=1}^k ...
1
vote
1answer
44 views

An arctan criterion for convergence?

Is the following inference correct, and if so, is it a mere curiosity? Let $\{a_k\}_{k \in \Bbb N}$ be a sequence of positive real numbers and set $$ P_k = \sum_{j=0}^{k-1} a_k \quad \mbox{and} ...
0
votes
0answers
40 views

Infinite sum of all natural numbers and solution of infinite sum of positive odds and evens [on hold]

The infinite sum of all natural numbers is believed to be -1/12. Now if you have this sum and you rearrange it so all evens are together and all odds are together the value should still be -1/12. Now ...
7
votes
1answer
108 views

Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
2
votes
3answers
70 views

Telescoping series: $\sum i^2 x^i$ for $0 < x < 1$

It is asked to find the sum $$ \sum_{i=1}^{\infty} i^2 x^i $$ Using the telescoping property. But I could not find a sequence to write my $s_n$ in function of and apply this.. Does anyone have a ...
0
votes
1answer
7 views

Sequence Interval of convergence

I could someone help me with the following sequence of functions of which I attempted to find the interval of convergence, but I couldn't get it to match with the solution I get from WolframAlpha ...
1
vote
3answers
59 views

Using Riemann sums to show that $\sum_{i=1}^n 1/i = \log{n} +c+O(1/n)$

I want to show that there exists a constant $c$ such that: $$ \sum_{i=1}^n 1/i = \log{n} +c+O(1/n) $$ I am thinking about Riemann sums. Any hints on that?
-1
votes
1answer
45 views

Find the next number in the series [on hold]

Can someone help me to find the next number of the following series 25,7,24,13,5,12,5,3,?
1
vote
1answer
16 views

Making a sequence alternating

So I have the sequence $\{x_{j}=k+\frac{1+2(j-\sum_{i=0}^{k}2^{n-i})}{2^{n-k}}\}_{j\in \mathbb{N}}$, where $n= \lfloor log_{2}j \rfloor$ and k$=\{l\in \mathbb{N}:$maximum l s.t. ...
3
votes
1answer
57 views

How should I prove that: $\sum_{i=1} ^{n}(\sin(\frac{i\pi}{n}))^2=\frac{n}{2}$

$$\sum_{i=1} ^{n}\Big(\sin\big(\frac{i\pi}{n}\big)\Big)^2=\frac{n}{2}$$ An interesting conclusion and checked for validity...holds for $n\geq 2$, but yet do not know how to prove it. Are there any ...
0
votes
1answer
32 views

how to prove using induction that sum of terms?

Prove that $\displaystyle\sum\limits_{i=1}^{k}\left(\dfrac{1}{(2i-1)}\dfrac{1}{(2i+1)}\right) = \dfrac{k}{(2k+1)}‎‎$ My Base of Induction is to check that it is true for i=1, so: ...
3
votes
0answers
56 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
1
vote
0answers
31 views

A question on infinite series and boundedness of sequence

Let $(a_n)$ be a real sequence such that for every convergent real series $\sum x_n$ of positive terms , $\sum |a_n|x_n$ is also convergent , then is it true that $(a_n)$ is a bounded sequence ?
0
votes
1answer
68 views

Closed form of solution of recurrence equation

Does there exists a closed form of solution of the following recurrence equation: $$a_{n+1} =a_n^2 -a_n +1$$
2
votes
1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
0
votes
1answer
51 views

find the sum of the series

If $a_1, a_2, \ldots, a_n$ are in arithmetic progression whose common difference is $d$,then find the sum: $$\sin(d) \cdot \left(\csc(a_1)\csc (a_2)+\csc(a_2)\csc (a_3)+\ldots+\csc(a_{n-1})\csc(a_n) ...