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)

2
votes
1answer
54 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
197 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
0answers
34 views

Find the next numbers in the sequence.. [on hold]

What is the next number in the sequence: 11, 67, 348, 1071, .... I tried factorizing and taking differences but I can't find common relationship in the sequence.
0
votes
1answer
38 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
29 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
40 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
108 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
45 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 ...
-3
votes
0answers
19 views

limits of a sequence inwhich all elements belonging to an open interval [on hold]

consider the interval $(-1,1)$ and a sequence $\{\alpha_n\}_{n=1}^{\infty}$ of elements in it. Then is every limit point of $\{\alpha_n\}$ is in $(-1,1)$ or is limit points of $\{\alpha_n\}$ can only ...
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
147 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
95 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
67 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}...
1
vote
0answers
25 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
84 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
54 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
41 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
votes
0answers
27 views

Sum Multi-Indexed Geometric Serie [on hold]

I am wondering whether it is possible to simplify the multi-indexed sum $$S(n) = \sum \limits_{k_{1}+2k_{2}+3k_{3}+...=n} T^{k_{1}}T^{k_{2}}T^{k_{3}}T^{k_{...}}.$$ Any suggestions will be appreciated....
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
53 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
65 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
112 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
43 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
99 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
43 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
31 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 ...
0
votes
0answers
14 views

How to find similar convergence rates?

Consider the Taylor's series infinite summation of $\sin(x)$. Let $A_k=\sum\limits_{i=0}^k(-1)^i{x^{2i+1}\over (2i+1)!}$ (Series expansion of $\sin(x)$) I need a series $\{C\}_n$such that its ...
0
votes
1answer
23 views

Cauchy sequence in $\mathbb{R}^d$

Is is possible to prove that sequence $\{x_n\}$ with terms in $\mathbb{R}^d$ has limit iff $\forall_{\epsilon > 0}\exists_{N \in \mathbb{N}}\forall_{n,m \ge N} \rho(x_n, x_m) < \epsilon$ as a ...
0
votes
0answers
48 views

Sum of a complex series involving $e^{2\pi i/N}$

I am trying to solve this sum of a complex series: $$\sum_{k=1, k≠\alpha}^N\frac{\beta^{(k-1)(\lambda-1)}}{\beta^{\alpha-k}-1}$$ where $\alpha,\lambda\in\{1,\ldots,N\}$, and $\beta:=e^{{2\pi i}/N}$. ...
1
vote
2answers
58 views

Does this sequence diverge $\frac{6^{3n}}{5^{4n}}$

I used the ratio test to get the problem in the following form: $\left|\frac{6^{3n+1}}{5^{4n+1}}\right| \cdot \left|\frac{5^{4n}}{6^{3n}}\right| = \frac{6}{5} \gt 1 $ so the sequence diverges?
2
votes
1answer
43 views

Proving monotonic decreasing of general term: $ \sum_{n=2}^\infty (-1)^{n+1}\dfrac{n^2-1}{n^3-1} $

I have got the following series - $$ \sum_{n=2}^\infty (-1)^{n+1}\dfrac{n^2-1}{n^3-1} $$ I know that this an alternating series which converge - but got confused on how to prove this the ...
4
votes
2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
0
votes
2answers
46 views

Is this a convergent sequence

$s_n = \frac{(-1)^n}{\sqrt[4]{n}}$ I'm using the ration test so I rewrote it like so: $\frac{(-1)^n}{(n+1)^{\frac{1}{4}}} \cdot \frac{n^{\frac{1}{4}}}{(-1)^n} =\lim_{n \to \infty}\frac{-n}{n+1}=-1$ ...
0
votes
2answers
55 views

Disprove a limit using epsilon definition

Disprove: If $\{x_n\}$ is any sequence, and $\{s_n\}$ is a sequence converging to $0$, then $$\underset{n \rightarrow \infty}{\lim} x_n s_n=0$$ I'm not sure how to go about solving this proof. I ...
0
votes
1answer
37 views

counting walks on a finite interval

Find the number of random walks on the integers $\{1, 2, ... m\}$ of length $n$. (For the purposes of this question a "random walk" is a sequence $\{a_i\}$ such that $a_i - a_{i+1} = \pm 1$.)
5
votes
3answers
80 views

Show that $a(n) = (1/n)^{1+1/n}$ is monotonically decreasing

$a(n)$ tends to $0$, as $n$ tends to $\infty$, but I am having trouble showing $a(n) > a(n+1)$. I tried to use ln (n+1) - ln n >= 1/(n+1). so ln (n+1) >= ln n +1/(n+1) => 1/(e^(ln n + 1/(n+1)) >= ...
1
vote
2answers
35 views

Show that the sequence is bounded?

Show that the sequence $\frac{3n + sin^2n}{6n}_{n \in \mathbb{N}}$ is bounded. I know that $ 0 \leq sin^2n \leq 1$, so $\frac{3n}{6n} = \frac{1}{2} \leq \frac{3n + sin^2n}{6n} \leq \frac{3n + 1}{6n}$ ...
0
votes
0answers
40 views

Questions Concerning “Approximate Polynomials”

In this paper, I encountered the following definition: Definition 2 (Approximate Polynomial) Let $U\subset \mathbb{C}$ and $\sigma\in\mathbb{N}\cup\{-\infty\}$. A function $f\colon U\to\...
2
votes
0answers
44 views

Understanding how to transform this infinite series into its closed form

On this page, there is an infinite series which the author simplifies "with a little bit of thought" hand-waving. I'm sure the author knew what he was doing but I cannot follow and I am hoping someone ...
0
votes
1answer
20 views

Right-const function and pointwise/uniform convergence

Let function $f:\mathbb{R} \rightarrow \mathbb{R}$ be right-const iff $\exists_{M \in \mathbb{R}}\forall_{x,y \ge M}f(x)=f(y)$. Consider function sequence $\{f_n\}$ which every term is right-const. ...
1
vote
1answer
44 views

Generalizing a Telescoping Sum $\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}$

I was trying to generalize an integral I found yesterday on this website and ran into the following interesting sum: $S_k=\sum_{n=1}^\infty \frac{1}{n+k}-\frac{1}{n}$. I have seen this sum come up a ...
1
vote
1answer
30 views

Taylor series expansion of function

I have the following statement - If $f$ has a Taylor series expansion about zero with radius $R$ , then $g(x) = \displaystyle f\left(\frac{x-1}{2}\right)$ has a Taylor expansion about $X = 1$ of ...
0
votes
4answers
54 views

How to prove this series converges ?

I have this series $\sum _{n=0}^{\infty }\:\left(\sqrt[n]{n}-1\right)^n$ Im having truoble to prove that this converges, I've tryind to use the ratio test but it didnt seem to get me to something ...
0
votes
0answers
40 views

Solution to a first order linear difference equation

The two questions are with respect to the following first order linear difference equation $(Y_{t} - Y_{t-1}) = (1-\lambda) (X_{t-1} - Y_{t-1})$, for $t \geq n$ Also, note that the process ...