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

learn more… | top users | synonyms (5)

0
votes
0answers
3 views

Sum of sequence of cubes and summation on the upper index

Express the sum of the sequence of cubes as a polynomial in n using the summation on the upper index formula: $$ \sum\limits_{k=0}^n\binom{k}{m} = \binom{n+1}{m+1} $$ It has been proven that the sum ...
2
votes
1answer
26 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
1
vote
0answers
26 views

Convergence of a series: TV show problem [duplicate]

I came across the following video on youtube where a kid was asked to show that $\sum_{n = 1}^{\infty} \frac{\sin{(2n)}}{1 + \cos^4{n}}$ is convergent. He tried to use the integral test but wasn't ...
13
votes
0answers
48 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
1
vote
0answers
28 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
0
votes
1answer
23 views

question involving power series for ln(x+a)

please could you help with this question. If a and b are small compared with x, show that $$ln(x+a) - lnx = \frac{a}{b}(1 + \frac{b-a}{2x})(ln(x+b) - lnx)$$ I've tried expanding ln(x+a) as a taylor ...
0
votes
1answer
34 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
1
vote
2answers
52 views

How can I prove that $\sum_{k=1}^{\infty}\frac{(2\sqrt[k]k-1)^k}{k^4}$ is convergent?

I'm trying to prove that the sum $\sum_{k=1}^{\infty}\frac{(2\sqrt[k]k-1)^k}{k^4}$ is convergent. I've tried Cauchy's root test - but I get the limit to be 1, so the test is inconclusive. I also ...
-1
votes
0answers
33 views

Power series for $\ln(1+x)$ and an estimate for $\ln(b/a)$ when $b\approx a$ [on hold]

I'm stuck on this question involving the power series for ln(1+x): $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$$ "Apply Maclaurin's series to establish a series for ln(1+x). ...
2
votes
2answers
37 views

Select a subsequence to obtain a convergent series.

Does there exists strictly increasing sequence $\{a_k\}_{k\in\mathbb N}\subset\mathbb N$, such that $$ \sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}}\lt \infty, $$ where $\delta>0$ given and ...
2
votes
1answer
155 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
4
votes
2answers
99 views

Convergent or divergent $\sum\limits_{n=0}^{\infty }{\frac{1\cdot 3\cdot 5…(2n-1)}{2\cdot 4\cdot 6…(2n+2)}}$

\begin{align} & \sum\limits_{n=0}^{\infty }{\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n+2)}} \\ & \text{ordering} \\ & a_{n}=\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot ...
1
vote
1answer
23 views

How to find the solution of integer equation group

I have the following problem: to find the general item of the following equation: let $a_1=b_1=1$, $$a_{n+1}=6a_n+2b_n, b_{n+1}=3a_n+2b_n$$ for any $n\geq 1$, find $a_n=?, b_n=?$
-1
votes
3answers
30 views

Determine whether this series converges using ratio test $\sum_{n=1}^{\infty}\prod_{j=1}^{n}\frac{3j-1}{4j-3}$ [on hold]

Umgh... I just have no idea, srsly, I've never done anything with sum of products
0
votes
4answers
30 views

how do I solve this arithmetic series

I have this arithmetic series $3+7+11+...+35+39$ to solve. So I see that there is a difference by 4 between the numbers and that there is a total of 9 terms. I plug these values in the following ...
4
votes
4answers
190 views

Does There Exist an Explicit Formula Describing Every Possible Sequence of Numbers?

This thread was previously titled "Does a Set Require an Explicit Formula to Exist?". I'm reading H. Enderton's Elements of Set Theory and working through understanding the Zermelo-Fraenkel axioms. ...
2
votes
1answer
36 views

Simplify a summation to reduce computation time

I am working on an optimization problem in which the following summation should be calculated in a computer program over a billion times. Therefore, I am looking for the possibility of somewhat ...
1
vote
1answer
37 views

test the convergence of an infinite series

How to prove that the $\displaystyle \sum_{n=1}^{+\infty} (1-e^{(-1/n^2)})$ series is convergent? I can not find a number to use the comparison test!
-1
votes
1answer
24 views

Proving that a sequence is Cauchy on the basis of squeeze theorem

Let $\{x_n\}$ be a sequence of real numbers such that $$|x_n| \leq \frac{2n^2 + 3}{n^3 + 5n^2 +3n + 1}$$ Prove $\{x_n\}$ is a Cauchy sequence Proof: Suppose that ${x_n}$ be a sequence of real ...
2
votes
1answer
42 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
0
votes
0answers
54 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
0
votes
1answer
29 views

summation of series with weird general term [on hold]

I'm having problem with this question: Find the sum of (n+1)2n+(n+2)(2n-1)+(n+3)(2n-2)+...+(2n-1)(n+2)+2n(n+1) ps: if possible, leave some advice for finding general term when dealing with summation ...
0
votes
0answers
30 views

What sequence should I use to visit color wheel? [on hold]

I am writing a program where I need to color N objects with unique colors. I have set up a function that maps the floating point value k [0,1] to a smoothly varying rainbow (say rgb=wheel(k)). If N ...
1
vote
2answers
52 views

Summation of series involving logarithm: $\sum (n+2)\ln 2^n$

The following question is: Show that $\sum\limits_{r = 1}^n {r(r + 2)} ={n \over 6}(n+1)(2n+7).$ Using this results, or otherwise, find, in terms of $n$, the sum of the series ...
1
vote
4answers
95 views

$\text{Prove that}$ $\frac{\sin(\frac{n+1}2)*\cos(\frac n2)}{\sin\frac 12} \ge\frac n2$

Prove that$$\frac{\sin\left(\frac{n+1}2\right)\times\cos\left(\frac n2\right)}{\sin\left(\frac 12\right)} \ge\frac n2$$ So far I've switched up the problem and gotten it down to all sin functions. I ...
0
votes
1answer
46 views

Is $n^3$ Bounded for $n\in\mathbb{N}$ as ${n\to\infty}$

Is $n^3$ Bounded for $n\in\mathbb{N}$ as ${n\to\infty}$ ? I am wondering if this is the case because I am thinking it bounded below by 1 but my textbook says its unbounded. Can someone explain why?
-1
votes
2answers
39 views

If $x_n \to +\infty$ then $(1+1/x_n)^{x_n} \to e$

I want to prove that if $x_n$ is a numerical sequence such that $\lim_n x_n=+\infty$ then $\lim_n (1+1/x_n)^{x_n}=e$. Should I pass by the continuous case (studying $f(x)=(1+1/x)^x$ for $x \to ...
-1
votes
2answers
40 views

What's the best way to find the sum of this sequence? [on hold]

I've got the following sequence: $$ 160-157+154-151+148-145+...+4-1 $$ How to find a sum of it?
6
votes
2answers
131 views

Does the series converge or diverge?

I want to check, whether $$\sum\limits_{n=0}^{\infty }{\frac{n!}{(a+1)(a+2)...(a+n)}}$$ converges or diverges. $a$ is a constant number Ratio test $$\begin{align} & ...
0
votes
1answer
39 views

If $\sum _{n=0 } ^{\infty } a _n < \infty$ $\sum_{n =0 } ^{\infty } b _n < \infty$ then $\sum_{n=0 } ^{\infty } a _nb _n < \infty$,

This seem obvious to be true but I'm unsure how to prove it or if there ara basic results about inifnite sums that apply. If $\sum _{n=0 } ^{\infty } a _n < \infty$ $\sum_{n =0 } ^{\infty } b _n ...
2
votes
2answers
35 views

summation of series (odd and even case)

Can anyone please answer this for me,it involves alternate signs which is different from normal summation formula. Q:Find the sum to n terms of the series $ 1^3 - 2^3 + 3^3 - 4^3 + \ldots -(n-2)^3 + ...
3
votes
2answers
121 views

Sums $\sum_{n=1}^{N}\sqrt{4n+1}$

I need to find sum of the first N terms of the sequence whose nth term is as follow : T(n)= $\sqrt{4*n+1}$ So the sequence is : $\sqrt{5}$,$\sqrt{9}$,$\sqrt{13}$,$\sqrt{17}$,$\sqrt{21}$...... ...
3
votes
3answers
82 views

How to prove $\sum\limits_{i=1}^{\infty}\dfrac{i}{2^i}$ converges? [on hold]

What would be the simplest way to prove that $\sum\limits_{i=1}^{\infty}\dfrac{i}{2^i}$ converges?
0
votes
1answer
24 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
4
votes
1answer
79 views

Cauchy Sequences, not converging to zero

True or False? If $\{x_n\}$ and $\{y_n\}$ are Cauchy and $x_n + y_n > 0$, for all $n\in\mathbb{N}$, then $\left\{\frac{1}{(x_n + y_n)}\right\}$ cannot converge to zero. I believe the claim to be ...
0
votes
0answers
30 views

Cauchy sequences, not converging to zero [duplicate]

True or False? If $\{x_n\}_{n\in\mathbb{N}}$ and $\{y_n\}_{n\in\mathbb{N}}$ are Cauchy and $x_n + y_n > 0$, for all natural $n$, then $$\left\{\frac{1}{x_n + y_n}\right\}_{n\in\mathbb{N}}$$ cannot ...
0
votes
1answer
39 views

alternating series $\sum(-1)^na_n$ is divergent, then, is $\sum A_k$ divergent?

An alternating series $\sum\limits_{n=1}^\infty (-1)^na_n$ is divergent , $a_n\geq0$, and $\lim\limits_{n\to\infty}a_n=0$. Could we conclude that $\sum\limits_{k=1}^\infty A_k$ is divergent, too ? ...
0
votes
0answers
21 views

Alternating root in a sequence

Is there a way to have a closed form for $P_{n}$, where $P_{1}=x$ $P_{2}=\sqrt{x}$ $P_{3}=x$ $P_{4}=\sqrt{x}$ $\vdots$
0
votes
0answers
34 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
2
votes
1answer
68 views

Sum of series with cosines

I need to prove this: $$ \sum_{n = 1}^{\infty}{8 \over \left(\,2n - 1\,\right)^{2}\pi^{2}}\, \sin\left(\,\left[\,2n - 1\,\right]\,{\pi x \over 2}\,\right) \sin\left(\,\left[\,2n - 1\,\right]\,{\pi z ...
0
votes
1answer
31 views

Finding $N$ when sum is given

$1^2 + 2^2 + 3^2 + 4^2 + \cdots + N^2 = S$ Given $S$ How to find $N$. The Formula to Find $S$ from $N$ is: $S = \frac{N(N+1)(2N+1)}6$ so this gives me a cubic equation: $2N^3 + 3N^2 + N = 6S$ ...
2
votes
1answer
72 views

Plouffe's formula for $\pi$

Plouffe established the following formula for $\pi$ in $2006$ $$\pi = 72\sum_{n = 1}^{\infty}\frac{1}{n(e^{n\pi} - 1)} - 96\sum_{n = 1}^{\infty}\frac{1}{n(e^{2n\pi} - 1)} + 24\sum_{n = ...
0
votes
0answers
30 views

Sum of roots of terms of an Arithmetic Progression

Is there any easy way or formula to calculate sum of roots of an Arithmetic progression? For example if the arithmetic progression is $a+d$,$a+2d$,$a+3d$, $\ldots$, $a+nd$: How can I calculate ...
1
vote
1answer
66 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
4
votes
3answers
465 views

Example of a divergent sequence

I want to produce a divergent sequence for which $|x_n - x_{n-1}| \to 0$. So far, I've only been able to show that $$\frac{x_n}{n} \to 0$$, which doesn't really help.
1
vote
2answers
43 views

Show that $\sum_{k=0}^{+\infty}ka_k=\sum_{k=0}^{+\infty}\sum_{i=k+1}^{+\infty}a_i$.

Let $(a_k)_{k\in\Bbb{N}}$ a family of positive real. Show that $$\sum_{k=0}^{+\infty}ka_k=\sum_{k=0}^{+\infty}\sum_{i=k+1}^{+\infty}a_i$$ So far, I have : By definition ...
1
vote
4answers
29 views

Function as a series :

Let $f(x)=\sum_{n=0}^{+\infty}\dfrac{x^n}{n!}$. Verify that $$\int_0^xf(x)dt=f(x)-1$$ This is the exercise 3 of the section $7.4$, of Guidorizzi's Calculus, Vol. 4. What I have tried: By the ratio ...
2
votes
2answers
75 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
2
votes
1answer
43 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
0
votes
1answer
32 views

Condition for derivative sequence to converge?

Let $E_n(R)$ be a sequence of function that converge to $E(R)$, When we can said that $\dot{E}_n(R)$ converge to $\dot{E}(R)$. I can assume: uniform converge of $E_n$ to $E$. $E(R)$ convex. ...