For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

learn more… | top users | synonyms (5)

0
votes
1answer
46 views

Fibonacci Sequence: Prove $f_1+f_3+\dots+f_{2n-1}=f_{2n}$ by Induction.

I believe the majority of my proof is correct I'm just not certain about the base case if any one can explain how to do that base case or fix any error I made I would greatly appreciate it. Recall ...
5
votes
0answers
84 views

Integral with an infinite sum

Let: $$\mathcal{S}(x)=\sum_{k=1}^{\infty}-\frac{\cos(\frac{\pi}{2}+k x)}{k^x}$$ I need help evaluating $$\int_{0}^{1}\mathcal{S}(x) dx$$ Obviously the cosine term in the numerator simplifies to $\...
1
vote
5answers
137 views

Sum of $1^2+3^2+\cdots+(2n+1)^2$ Have trouble with proof.

I've been working through a question on Courant's What is Mathematics? This is the question: Prove $1^2+3^2+\cdots+(2n+1)^2=\frac{n(n+1)(2n+1)(2n+3)}{3}$. I called this $S_{(2n+1)^2}$. What I've ...
2
votes
2answers
24 views

Is the Cesàro Sumation of Series divergent to infinity divergent?

More specifically the question is: If I have a series $(u_n)_{n\in\mathbb{N}} \subset \mathbb{R}$ that diverges to infinity. Then it's cesaro sumation series $(s_n)_{n\in\mathbb{N}}=(\frac{1}{n+1}\...
2
votes
0answers
35 views

Why are sequences and functions notated differently?

Why do we usually write, e.g., $s_n$ for sequences, while functions are usually written as $f(x)$? Conceptually, aren't sequences just functions with a subset of the naturals, not of the reals, as ...
3
votes
5answers
130 views

Does the series $\sum_{n=1}^\infty\frac{n^{\sqrt{n}}}{n!}$ converge?

Immediately I recognize that there's a factorial and I use the ratio test to try and solve it: $$\lim_{n \rightarrow \infty}\left|\frac{{(n+1)}^{\sqrt{n+1}}}{(n+1)!}\cdot\frac{n!}{n^{\sqrt{n}}}\right|...
0
votes
0answers
33 views

Continous functions and zeros

How to prove following theorem? If sequence $\{f_n\}$ of continous real functions with domain $D \subset \mathbb{R}$ is compact convergent to $f$ and sequence $\{x_n\}$ with $D$ satisfies $f_n(x_n) = ...
1
vote
1answer
38 views

I am trying to prove the distribution function for the 'birthday problem' can anyone help?

Let $Y_1, Y_2, . . .$ be i.i.d. and uniformly distributed on the set ${1, 2, . . . , n}$. Define $X^{(n)} = \min \{k : Y_k = Y_j \,\,for \,\,some \,\,j < k\} $, the first time that we see a ...
0
votes
1answer
41 views

Variant of the geometric series

Can someone explain me how one computes $$\sum_{k=1}^n kq^k = \dfrac{nq^{n+2}-(n+1)q^{n+1}+q}{(1-q)^2}$$ and what exactly the derivative has to do with it?
0
votes
0answers
55 views

Find the limit of sequence.

Assume $ f(x) \in C^2[a,b] $ and $f(a)f(b)<0,f'(x)>0,f''(x)>0,\forall x \in[a,b]. $Prove the sequence $$ x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n) } \quad x_1 \in[a,b]\text{ and } x_1 \text{is to ...
5
votes
5answers
112 views

Bounding a series: $\frac{\pi}{2} < \sum_{n=0}^\infty \frac{1}{n^2 + 1} < \frac{3\pi}{2} $

I have the following statement - $$\frac{\pi}{2} < \sum_{n=0}^\infty \dfrac{1}{n^2 + 1} < \frac{3\pi}{2} $$ So I tried to prove this statement using the integral test and successfully proved ...
1
vote
0answers
33 views

Binomial Series Expansion and then find an approximate value [closed]

This is from an A-Level Maths paper. Show that $\frac{x}{(1-x)^3} = x + 3x^2 + 6x^3 + O(x^4)$ (Only first three terms of infinite series expansion are asked for) Use the result to find an approximate ...
2
votes
1answer
40 views

Where is the mistake of a possible application of Frullani's theorem in this case?

My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ...
0
votes
2answers
58 views

What is the relation between $x,y$ if $\tan(20^\circ),x,\tan(50^\circ)$ and $\tan(20^\circ),y,\tan(70^\circ)$ are in AP?

If $\tan(20°),x,\tan(50°)$ are in AP and $\tan(20°),y,\tan(70°)$ are in AP then relation between x and y is?. $$\text{Attempt}$$. As they are in AP So $2x=\tan(20°)+\tan(50°),2y=\tan(20°)+\tan(70°)$ ...
-5
votes
1answer
119 views

Telescoping function Revealed.

Part B: I found Summation $$\sum_{n=0}^\infty\frac{x^n \ (-1)^n}{(y+1)^{n+1}} = \frac{1}{(x+y+1)}$$ However $x$ is related to $y$. $y \ge |x|$. You may derive the result and present it by ...
1
vote
0answers
58 views

Limit of $\sum_{k=0}^{n}\frac{1}{2k+n}$ and similar

Examine wether following sequences have limits and if yes - find them. a)$\sum_{k=0}^{n}\frac{1}{2k+n}$ b)$\sum_{k=0}^{n}\frac{(-1)^n}{2k+n}$ c)$\sum_{k=0}^{n}\frac{(-1)^k}{2k+n}(\frac{1}{3})^k$ a)...
4
votes
1answer
174 views

New series formula for $\arctan(x)$?

I discovered this equation, but have no idea if it has been previously discovered. Please help determine if it has been previously developed. Or please prove that the equation is not correct. $$\...
0
votes
2answers
43 views

How to show the sequence is monotone

"$u_n = \frac{2}{1+e^{-n}}$. Show that $u_n$ is monotone." My approach would be to consider |$u_{n+1} - u_n$| = |$\frac{2}{1+e^{-n-1}} - \frac{2}{1+e^{-n}}$|. However I'm not sure the best way to ...
2
votes
3answers
53 views

prove that $\sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)}$ is convergent and find the limit when $n \to \infty$

does the following sumatory converges? if yes find the limt when $n \to \infty$ $$\sum_{n=1}^\infty \frac{2^n+n^2+n}{2^{n+1}n(n+1)}$$ ideas? i have tried by the comparison test.
2
votes
1answer
28 views

Greatest number in fibonacci sequence with property: sum of digits=index in fibonacci sequence

I came across very interesting sequence based on fibonacci sequence. From fibonacci numbers we choose only elements with digit sum=index in fibonacci sequence. It is very interesting that we most ...
19
votes
5answers
3k views

A very curious rational fraction that converges. What is the value?

Is there any closed form for the following limit? Define the sequence $$ \begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$ with initial values $a_0 = b_0 = 1$. ...
2
votes
1answer
60 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
4
votes
4answers
201 views

How to prove that the series $\sum\limits_{n=1}^\infty {\sin^2n} $ diverges

I want to use a divergence test to prove that $\lim_{n\to \infty} \sin^2n$ does not converge. So $\sum_{i=1}^\infty \sin^2 n $ diverge. But because $\pi$ is an irrational number. So I cannot use ...
0
votes
1answer
39 views

$\succsim$ preorder on X being continuous imply lower contour set closed

$\succsim$ is preorder (i.e. preference relation) on X that is continuous. This implies the lower contour set is closed. Would you please share your 2 cent on my parenthesis explanation (e.g. line ...
0
votes
3answers
30 views

bounds on a sequence

It may look that this question is trivial, but: Let $(a_n)_{n=1}^\infty$ a sequence s.t. $\forall n\in \mathbb{N} \ \ a_n<\frac {1}{n}$. Prove/Disprove : there is $c > 1$ s.t. $\forall n\in \...
1
vote
1answer
47 views

Limit of the fraction $\dfrac{n(n+1)^\alpha}{\sum_{k=1}^nk^\alpha}$

I'm stuck in calculating the following limit: $$L=\displaystyle\lim_{n\to\infty}\dfrac{n(n+1)^\alpha}{\sum_{k=1}^nk^\alpha}$$ For what values of $\alpha\in\mathbb{R}$ $L$ has a finite value? Thanks.
0
votes
1answer
42 views

Partial sums of periodic sequences

Let $a_i$,$b_i$ be two periodic real sequences with a period of $n$. For $k\leq n$, denote the $k$-length partial-sums starting at $j$ by $a[j:k],b[j:k]$, i.e: $$a[j:k] = \sum_{i=j}^{j+k-1}a_i\,\,\,\,\...
3
votes
1answer
109 views

Finding all $z\in \mathbb{C}$ such that the series $\sum\limits_{n=1}^{\infty} \frac{1}{1+z^n}$ converges

I am trying to find out all $z\in \mathbb{C}$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+z^n}$ converges. I notice that for $\left|z\right|\leq 1$, we have $\left|1+z^n\right|...
1
vote
1answer
47 views

derivation of fibonacci log(n) time sequence

I was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the ...
4
votes
4answers
64 views

Convergence in $\mathbb{Q}$

How will I prove that a sequence in $\mathbb{Q}$ which is bounded below and decreasing is Cauchy, without using the knowledge of reals?
1
vote
1answer
25 views

Prove, using the definition, that $(2 + \frac{1}{n^2})$ is a Cauchy sequence.

"Prove, using the definition, that $(a_n)_{n \in \mathbb{N}} = (2 + \frac{1}{n^2})$ is a Cauchy sequence." My answer: Let $\epsilon > 0$ and choose $N = \frac{1}{\epsilon}$. Then for all $n, m \...
3
votes
2answers
153 views

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...
1
vote
1answer
47 views

A convergent series implies two split convergent series

Consider the series $$ \sum_{n=0}^{\infty}\left ( \frac{1}{n+1}-\frac{1}{z+n} \right )\tag{1} $$ It converges for all $z\notin \{0\}\cup \mathbb{Z}^-$. Does it imply, that $$ \sum_{n=0}^{\infty}\frac{...
6
votes
3answers
100 views

A common term for $a_n=\begin{cases} 2a_{n-1} & \text{if } n\ \text{ is even, }\\ 2a_{n-1}+1 & \text{if } n\ \text{ is odd. } \end{cases}$

When I was answering a question here, I found a sequence as a recursive one as given below. $a_1=1$, and for $n>1$, $$a_n=\begin{cases} 2a_{n-1} & \text{if } n\ \text{ is even, }\\ 2a_{n-1}+...
2
votes
0answers
68 views

Question about an infinite product

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
2
votes
1answer
67 views

Explaining a Series Expansion for the Divisor Function

The "sum-of-divisors" function, defined as $\sigma(n)=\sum_{d\,\mid\,n}d$, can be expressed as $$\sigma(n)=\sum_{i=1}^n\sum_{j=1}^i\cos\Big(2\pi n \frac{j}{i}\Big)$$This expansion makes logical sense. ...
0
votes
2answers
87 views

Riemann zeta function functional equation proof explanation

In Riemann zeta function functional equation proof I arrived to a following equation $$\frac{\Gamma\left(\frac{s}2\right) \zeta(s)}{\pi^{\frac{s}2}}=\sum_{n=1}^\infty \int_0^\infty x^{\frac{s}2-1}e^{-...
1
vote
2answers
55 views

How to find the Summation of series of Factorials?

$$1\cdot1!+2\cdot2!+\cdots+x\cdot x! = (x+1)!−1$$ I don't understand what's happening here. The given sum of factorials is generalized into a single term. Could somebody please help me finding the ...
0
votes
3answers
42 views

Does this sequence have a closed form representation?

We know that $$ \sum_{s=0}^\infty \frac{\lambda^{s}}{s!} = e^\lambda$$ Relatedly, $$ \sum_{s=1}^\infty \frac{\lambda^{s}}{s!}s = \lambda \sum_{s=1}^\infty \frac{\lambda^{s-1}}{(s-1)!}$$ For which ...
1
vote
0answers
23 views

Convergence of $\sum_{n=1}^\infty\frac{\psi(n)}{e^n}\sin ns$ on an horizontal closed strip

Let $\psi(x)=\sum_{k\leq x}\Lambda(k)$ the Second Chebyshev function, and $\epsilon>0$. I would like to ask Question. Can you prove or disprove that the series $$\sum_{n=1}^\infty\frac{\psi(n)}{...
0
votes
2answers
48 views

The limit of general term in a series

I have the following statement - If $\sum_{1}^{\infty} a_{n}^2$ converge then $\sum_{1}^{\infty} a_{n}^3$ converge. Well i know this statement is true , but if can someone explain why $\lim_{n\...
0
votes
0answers
56 views

Polynomial Sequences

I recently encountered the following definition: there exists a sequence of polynomials $(p_n)_{n\in\mathbb{N}}$ of fixed degree $\sigma$ such that for every $x\in U$, $\left|f(n+x) - p_n(n+...
4
votes
2answers
66 views

$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2+n}$

Does the following series converge or diverge $$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2+n}$$ I know that $\sum_{n=1}^{\infty}\frac{1}{(\ln{n})^2}$ diverges. $\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2}$ ...
12
votes
1answer
114 views

The divergent sum of alternating factorials

So I came across this exposition of a paper by Euler here where Euler is trying to sum the divergent sum: $$s = 1 - 1 + 2! - 3! + 4! \dots = \sum_{k\geq 0}(-1)^k k!.$$ There are a couple of questions ...
2
votes
0answers
46 views

An infinite product for $\arccos \alpha$ similar to Viete's formula for $\pi$

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
0
votes
0answers
32 views

Function sequence and some properties

Consider functions $f_n$, $f : \mathbb{R} \rightarrow \mathbb{R}$ such that the sequence $\{f_n\}$ is uniformly convergent to $f$ and every $f_n$ has property $W$. Determine whether $f$ must have $W$ ...
1
vote
2answers
44 views

Existence of positive integer solution of a equation

I'm trying to find if the following equation has positive integer solutions $$x + (x+y) + (x+2y) + (x+3y) + \cdots + (x+(n-1)y) = z$$ where $z$ and $n$ are given. I can't progress further. -> $xn +...
3
votes
2answers
100 views

Sum of $\frac{1}{5}-\frac{2}{5^2}+\frac{3}{5^3}-\frac{4}{5^4}+…$

Find the sum of following series: $S=\frac{1}{5}-\frac{2}{5^2}+\frac{3}{5^3}-\frac{4}{5^4}+....$ upto infinite terms Could someone give me slight hint to solve this question?
0
votes
1answer
47 views

Validity of Arithmetic Progression

Given the sum of arithmetic progression and number of terms . We have to determine whether the arithmetic progression exists or not . First term and common difference should be natural numbers . e.g -...
1
vote
0answers
32 views

Riemann series theorem wikipedia example problem

In the wikipedia article of Riemann series theorem , they give us an example of Riemann series theorem by calculating the alternated harmonic sum that converge to ln(2). And they show that by ...