Tagged Questions

Recurrence relations, convergence tests, identifying sequences

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0
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0answers
19 views

Cauchy product on power series

Original posting by dioxen here: Double summation including power and factorial I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac ...
-1
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1answer
58 views

Formula for series $\frac{\sqrt{a}}{b}+\frac{\sqrt{a+\sqrt{a}}}{b}+\cdots+\frac{\sqrt{a+\sqrt{a+\sqrt{\cdots+\sqrt{a}}}}}{b}$

All variables are positive integers. For: $$a_1\qquad\frac{\sqrt{x}}{y}$$ $$a_2\qquad\frac{\sqrt{x\!+\!\sqrt{x}}}{y}$$ $$\cdots$$ ...
10
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1answer
80 views

How to prove $\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$

One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$ With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin ...
3
votes
1answer
63 views

Evaluating a summation of inverse squares over odd indices

$$ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ I want to evaluate this sum when $n$ takes only odd values.
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4answers
45 views

Telescoping sum of powers

http://i.stack.imgur.com/wiVEH.png Can somebody explain me how these results are disposed intuitively? I didn't understand why $$(n-1)^3 -(n-2)^3$$ became equals to $$3(n-1)^2 - 3(n-1) + 1$$ How do ...
1
vote
1answer
54 views

Prove product of uniformly convergent sequences of functions is the product of the limiting functions [duplicate]

The question is "Prove that if $f_n(x)\to f(x)$ uniformly and $g_n(x)\to g(x)$ uniformly,both in [a,b] $f_n(x)g_n(x)\to f(x)g(x)$ uniformly. What I tried we are given that $\forall \epsilon ...
1
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1answer
28 views

Summation with factorial terms (involving Laguerre polynomials)

As part of an exercise including gamma functions and Laguerre polynomials, I need to show that for a Laguerre polynomial $L_n(x)$, $$\int\limits_0^\infty L_n(x)x^ke^{-x}dx = 0 \textrm{ with } n \in ...
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4answers
93 views

How is “n+n/2+n/4…1” equal to “2n-1” using the formula for geometric series?

I never knew not having good knowledge of basic maths will be so crippling!! So please help me out this time. I'll be working on my maths from today on. I was discussing about complexity of an ...
1
vote
2answers
55 views

Counterexample to inverse Leibniz alternating series test

The alternating series test is a sufficient condition for the convergence of a numerical series. I am searching for a counterexample for its inverse: i.e. a series (alternating, of course) which ...
1
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3answers
39 views

Uniform convergance for $f_n(x)=x^n-x^{2n}$

the function $f_n(x)=x^n-x^{2n}$ converge to $f(x)=0$ in $(-1,1]$. Intuativly the function does not converge uniformally in (-1,1]. How can I prove it? I tried using the definition $\lim ...
4
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1answer
39 views

trigonometric identity related to $ \sum_{n=1}^{\infty}\frac{\sin(n)\sin(n^{2})}{\sqrt{n}} $

As homework I was given the following series to check for convergence: $ \displaystyle \sum_{n=1}^{\infty}\dfrac{\sin(n)\sin(n^{2})}{\sqrt{n}} $ and the tip was "use the appropriate identity". ...
0
votes
3answers
33 views

Example of series of the form $\sum \frac{1}{n^{1+a_n}}$

How one can find a sequence $(a_n)$ of positive numbers such that $\lim a_n=0$ and $$\sum^{\infty}_{n=i} \frac{1}{n^{1+a_n}}$$ converges. Thank you!
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votes
3answers
50 views

Sequences with the following properties…

Suppose that $\lim_{n\to\infty}\frac{a_n}{b_n}$=1. And $\sum^{\infty}_{n=1}a_n$ converges, $\sum^{\infty}_{n=1}b_n$ diverges. Are such sequences $(a_n)$, $(b_n)$ exist?
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2answers
132 views

The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$

What is the answer to the following limit of a power series? $$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
5
votes
2answers
89 views

Calculate the limit of two interrelated sequences?

I'm given two sequences: $$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$ as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle ...
1
vote
1answer
48 views

What programs can calculate (this) series (get expression in closed form)?

What programs can calculate this (type of) series? $$ \sum_{m,n=0}^\infty \frac{(-1)^{-2n}2^{-2m-n}\,(1+m)}{2+m+n} $$ One program I know of is XSummer math-ph/0508008, XSummer. The result is: $$ ...
9
votes
1answer
111 views

Can this sum ever converge?

If I have a strictly increasing sequence of positive integers, $n_1<n_2<\cdots$, can the following sum converge? $$ \sum_{i=1}^\infty \frac{1}{n_i} (n_{i+1}-n_{i}) $$ I suspect (and would like ...
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1answer
29 views

Extracting monotone distinct sequence

Let $(x_n)$ be a sequence in $\mathbb{R}$ and $x_n\rightarrow p$ for some $p\in \mathbb{R}$. Suppose $x_n=p$ for finitely many index. Then there is a strictly monotone distinct (every element is ...
3
votes
1answer
69 views

Infinite sum convergence $ \sum_{i\geq 1}\frac{1}{x^i-y^i}$

For certain values of x and y, the sum $$\sum_{i=1}^{\infty}{\frac{1}{x^i-y^i}}$$ converges...is there a way to get the exact value, given x and y?
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0answers
23 views

Expressing a Sequence as a Function of n (Cartan Groups)

The problem is concerning a variation of A141419, the only difference is that my sequence, instead of being like shown on OEIS: {1}, {2, 3}, {3, 5, 6}, {4, 7, 9, 10}, {5, 9, 12, 14, 15}, {6, 11, 15, ...
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vote
3answers
47 views

Which one is the correct series expansion?

Is $$p^{n+1} = p^0+p^1+ \dots + p^n$$ or $$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$ I am confused. please explain the correct one.
1
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1answer
31 views

Alternating series - absolute convergence?

The question is, whether the following series is convergent, absolutely convergent or divergent $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}.$$ The term $\dfrac{(-1)^{n+1}}{2n-1}$ = $ ...
0
votes
3answers
52 views

Can someone explain to me what are these 2 statements talking about?

I have to prove that these 2 statements are equivalent, but I can't even understand them. There exist $\epsilon_0>0$ such that for all $k\in\mathbb N$, there exist $n_k\in\mathbb N$ such that ...
0
votes
1answer
17 views

Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?

I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a ...
1
vote
1answer
28 views

Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_n)=0$.

Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_{n_k})=0$. I guess I have to prove that $(x_{n_k})$ diverge, but I don't know how to ...
2
votes
1answer
113 views

Need help to proof

I got the result below during my research. $$1=\frac{1}{1+a_1}+\frac{a_1}{(1+a_1)(1+a_2)}+\frac{a_1a_2}{(1+a_1)(1+a_2)(1+a_3)}+\frac{a_1a_2a_3}{(1+a_1)(1+a_2)(1+a_3)(1+a_4)}+... \tag 1$$ ...
1
vote
4answers
78 views

Is $\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k$ always even?

Is $$ f(n,x,y)=\sum^{n-1}_{k=1}{n\choose k}x^{n-k}y^k,\qquad\qquad\forall~n>0~\text{and}~x,y\in\mathbb{Z}$$ always divisible by $2$?
3
votes
2answers
152 views

Is there an expression for the ith term of this sequence $1,1,2,1,2,3,1,2,3,4,1,2,3,4,5…?$

I'm trying to do some work in Excel and if I found a formula for this sequence it would help a lot. I don't particularly need to know why the formula works. I have found the sequence here . But ...
1
vote
1answer
22 views

general formula for partial sum of series

im having trouble figuring out how to find the general formula for partial sums of a series. Is it a trial and error kind of thing where I just have to figure it out? or is there a systematic way to ...
1
vote
2answers
53 views

Help me solve this recurrence relation

I'm trying to solve the recurrence relation $$a_n = (\lambda +\mu)a_{n+2}+\mu a_{n+3}.$$ with initial relationships of: $\lambda a_1 = \mu a_2$ $(\lambda +\mu)a_2 = \mu a_3.$ I found a site ...
1
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1answer
42 views

REVISTED$^1$ - Order: Modular Arithmetic

Relevant Literature: Question: Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$. Thoughts: Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
0
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2answers
47 views

Prove the convergence of the series. [duplicate]

Let r > 1 be a real number. Prove that the following series is convergent. $$\sum_{n = 1}^{\infty}\frac{1}{n^r}$$
1
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1answer
45 views

$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then

$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then 1.$\sum f_n$ converges uniformly on $D$? 2.$f_n$ and $f'_n$ converges uniformly on $D$? 3.$\sum f'_n$ converges on $D$ pointwise? 4.$f_n''(z)$ ...
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0answers
12 views

test of sheffer polynomials

Are the any tools available for testing if a polynomial sequence corresponds to a generating function (e.g Sheffer sequence GF)? The Coxeter configuration is a configuration whose incidence graph is ...
3
votes
1answer
49 views

Question about Euler numbers

How to prove that where $\ E_{2n}$ is the $2n^{th}$ euler number and $$\frac{1}{\cosh(x)}=\sum_{n=0}^{\infty }\frac{E_n}{n!}x^n$$ is there any help? thank for all
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1answer
25 views

the relationship between generating function and recurrence

Does the existence of a generating function of a sequence imply the existence of recurrence relation of the same sequence?
0
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0answers
29 views

Are these series convergent?

I came across the following two series while trying to solve Laplace's equation in two dimensions. $$T_1 = \sum_{n=1}^{\infty}\rho^n(A_n\cos(n\phi) + B_n \sin(n\phi))$$ $$T_2 = ...
4
votes
2answers
88 views

$\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$

Let $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$,$s_k$ denotes the partial sum then could anyone tell me which of the following is/are correct? $1.$ $s_k=0$ for ...
2
votes
2answers
39 views

Find an interval of convergence and an explicit formula for $f(x)$

Let $f(x) = 1 + 2x + x^2 + 2x^3 +x^4+...$ If $c_{2n} = 1$ and $c_{2n+1} = 2$ $\forall n \ge 0$ find the interval of convergence and an explicit formula for $f(x)$. The answers are $I = (-1,1)$ and ...
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0answers
36 views
+150

Mapping Between Sequences: Example

Take $0\leq r < j$, and let all values be nonnegative and integer. Consider the function on a sequence ${x(n)}$, $\Phi_jx(mn+r)=mx(n)+\frac{r}{m}(x(n+1)-x(n))$, where we consider $x(0)=0$. As an ...
11
votes
2answers
208 views

An infinite product

Given that $0 < a < 1$, what is the value of $$ P = \lim_{n\to \infty} \prod_{i=1}^n \frac{1 - (2i + 1)a/(2n)}{1 - ia/n} $$ Thus, $P_n$, the $n$th term, is $$ P_n = \frac{1-3a/2n}{1-a/n}\cdot ...
1
vote
1answer
36 views

Series of Vectors

In $\mathbb{R}^n$ we define sequences of it's elements in a very natural manner, we say that a sequence is a function $x : \mathbb{N} \to \mathbb{R}^n$ and we denote it by $(x_k)$ as in the $n=1$ ...
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2answers
19 views

Question on AP(sequences and series)

Prove that sqrt(2), sqrt(3) and sqrt(5) cannot be terms of an A.P.(not necessarily consecutive)!
2
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1answer
42 views

Calculation Of Integral Related To Sequence

Let's evaluate the following integral. Many trials but no success. $$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$$
0
votes
1answer
30 views

How to represent a sequence of odd numbers given specific criterea

I'm trying to figure out how to represent a sequence of ODD numbers given the following conditions: 1) I know how many numbers are in the sequence (N). 2) I know the average of all the numbers in ...
1
vote
1answer
44 views

Taylor series expansion and approximation

I found this amazing question in the last calculus exam, but I don't know how to answer. Let $T(x) = \ln(1+a) + \frac{1}{1+a}(x-a) - \frac{1}{2(1+a)^2}(x-a)^2 +...+ ...
7
votes
1answer
103 views

convergence of series with $k!$

check if the following series converges: $\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$ where $k!!=k(k-2)(k-4)(k-6)...$ I came across this exercise while going trough some old exams. I'm ...
1
vote
1answer
27 views

identity of polylogarithm

let be the function defined by a series $$ f(x)= \sum_{n=0}^{\infty}g(n)x^{n} $$ assume also that $ g(n)= \sum_{k=0}^{\infty}a(k)n^{k} $ then we have the double series $$ f(x)= ...
3
votes
3answers
104 views

Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$

Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that $$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$ ? I think it is but I can't prove it. Of course if $a_n ...

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