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

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7
votes
3answers
721 views

Evaluating $\sqrt{6+\sqrt{6+\cdots}}$

Tough as introduction to analysis for beginners (Dutch handbook - I'm Belgian). Again ($n$) means index $n$, $x_1 = \sqrt6$, $x_{n+1} = \sqrt{6+x_n}$ Question: $$|x_{n+1} - 3| \le 1/5 \cdot |x_n - ...
1
vote
0answers
290 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
0
votes
1answer
77 views

Sequence of $n$ terms, generated for a given $k$-subset sum

Before I make myself clear, I wish to say that I do not want to know a sequence like the binary sequence. If possible, I'd want a sequence smaller than the binary sequence, of polynomial growth rate ...
1
vote
0answers
325 views

Unclear on an algebra step in Laurent series of cotangent.

I'm reading through a calculation for $\displaystyle z\pi\cot(\pi z)=\pi iz+\frac{2\pi iz}{e^{2\pi iz}-1}$ which confuses me. It states $$ \begin{align*} z\pi\cot(\pi z) &= \pi iz+\frac{2\pi ...
1
vote
2answers
76 views

Convergence of $\sum_{n=1}^\infty (3x+1)^{-3n}$

Can anyone think of an easy way to find the values of $x$ for which the following series converges? $$\sum_{n=1}^\infty (3x+1)^{-3n}$$ I'm thinking some convergence test (root test, perhaps?) but ...
0
votes
1answer
78 views

Can someone help what test to use in this equation to find the Radius of convergence and the interval of convergence?

Can someone help what test to use in this equation to find the Radius of convergence and the interval of convergence? $\displaystyle\sum (-1)^n\frac{(x+2)^n}{n}$
3
votes
2answers
313 views

Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges.

I have been working on this problem for a while and cannot seem to make any progress without coming up with something wrong or hitting a dead end. Here is what I have so far: $ \prod (1+a_n) < ...
3
votes
2answers
94 views

$|\sum |a_n| - |a_n|^2|$ is bounded and $\sum|a_n|^2$ converges , what can be concluded about $\sum a_n$?

If I have that $|\sum |a_n| - |a_n|^2|$ is bounded and know that $\sum|a_n|^2$ converges can I conclude anything about $\sum a_n$?
3
votes
2answers
923 views

Proving that if the partial sums are bounded (i.e. $|\sum_n^N a_n|$ is bounded) and that $\sum |a_n|^2$ converges, then $\sum a_n$ converges.

I am wondering if the following is true and if so need help proving it. If the series of partial sums is bounded, that is $|\sum_{n=1}^N a_n|$ is a bounded sequence indexed by $N$ and that ...
1
vote
3answers
86 views

Show that $(X, d_2)$ is incomplete

I have a set $X = [0, \infty)$ and two metrics: $$ d_1(x, y) = |x-y| $$ $$ d_2(x, y) = \left| \frac{x}{1+x} - \frac{y}{1+y} \right| $$ I already showed that $d_1$ is equivalent to $d_2$. Now I have ...
2
votes
2answers
202 views

How to expand undifferential function as power series?

If a function has infinite order derivative at $0$ and $\lim_{n\to \infty}(f(x)-\sum_{i=1}^{n} a_{n}x^n)=0$ for every $x \in (-r,r)$,then it can be expand as power series$\sum a_{n}x^n$, My question ...
3
votes
2answers
216 views

Question about sum

A friend of mine asked me to help with this problem. I tried induction, but I didn't know how to get this formula. If $x$ and $y$ are real numbers such that $xy= ax+by$. Show that $$ ...
6
votes
5answers
417 views

Positive series problem

Let $\sum\limits_{n\geq1}a_n$ be a positive series, and $\sum\limits_{n\geq1}a_n=+\infty$, prove that: $$\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty.$$
1
vote
0answers
66 views

Series of $\sum_{i=0}^k 2^{n/{2^i}}$

$$\sum_{i=0}^k 2^{n/{2^i}}$$ I'm trying to find an actual sum of this little nice sum , but I think that there's a problem with it being a geometric series . I'd appreciate any help Regards
0
votes
6answers
297 views

Obtain the formula for the following sequence

I can't seem to figure out how to find an algebraic formula for the following sequence of numbers. $$0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0$$ Can somebody ...
0
votes
1answer
179 views

Evaluation of $\sum_{x=1}^{\infty}x^{-x}$ [duplicate]

Possible Duplicate: “Closed” form for $\sum \frac{1}{n^n}$ Is it possible to evaluate this sum, and if so, how would you do it? This question has been irritating me for a ...
2
votes
1answer
83 views

Limits of sequence $(1 + \frac{3}{n^2})^{n^2}$ as n tends to infinity

I need to find $\lim_{n \to \infty}$ $(1 + \frac{3}{n^2})^{n^2}$ and I've been given the following: $\lim_{n \to \infty}$ $n^{1/n}$ = 1, $\lim_{n \to \infty}$ $a^{1/n}$ = 1 and $\lim_{n \to \infty}$ ...
3
votes
4answers
217 views

How to compute $\sum\limits_{n=3}^{\infty}\frac{(n-3)!}{(n+2)!}$

I came across a question which required us to find $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n^5-5n^3+4n}$. I simplified it to $\displaystyle\sum_{n=3}^{\infty}\frac{1}{(n-2)(n-1)n(n+1)(n+2)}$ which ...
1
vote
1answer
78 views

Last (For Now) Question On Limit Evaluation

What (if exists) the $\lim \limits_{n\to \infty}\frac{(n+1)^n-(n-1)^n}{n^n+2}$? Should I use the binomial theory in the numerator? Please try to keep it as elementary as possible because we are only ...
1
vote
2answers
375 views

what is value of x in an arithmetic progression involving logarthms

$\log 2,\log 2^{x-1}$, and $\log 2^{x+3}$ are $3$ consecutive terms of an arithmetic progression; find (i) the value of $x$;
5
votes
3answers
378 views

What Is The Limit Of The Sequence: $\frac{n^3}{{((3n)!)^\frac{1}{n}}}$

What (if exists) the $\lim \limits_{n\to \infty}\dfrac{n^3}{{((3n)!)^\frac{1}{n}}}$? I have no idea where to begin. Maybe I could use the ratio test? Please try to keep it as elementary as possible ...
2
votes
3answers
2k views

Is there a shortcut for calculating summations such as this? [duplicate]

Possible Duplicate: Computing $\sum_{i=1}^{n}i^{k}(n+1-i)$ I'm curious in knowing if there's an easier way for calculating a summation such as $\sum_{r=1}^nr(n+1-r)$ I know the summation ...
4
votes
3answers
2k views

Matrix exponential convergence

Help me please to prove that matrix exponential which defined as: $e^{A}=\sum\limits_{k=0}^{\infty }\frac{A^{k}}{k!}$ converges for all matrix $A$ Thanks beforehand.
1
vote
1answer
69 views

Approximation to a product of sequence $\prod_{i=0}^{n}(1-\sigma i)$ when $\sigma$ is very small

Is there a formula I can use as an approximation to the following equation for velocity of a projectile when $\sigma$ is very small? $$\dot{x}[n] =v_0\prod_{i=0}^{n}(1-\sigma i)$$ ...
27
votes
2answers
2k views

Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
0
votes
2answers
157 views

computation of sum with natural numbers and roots

Let $n_1,...,n_m$ be non-negative natural numbers, such that their sum $n_1+\dots+n_m=2n$. Fix $m$. I am wondering how to compute, or bound from above $$\sum_{i=1}^m \sqrt{1+\frac{1}{2n_i-1}}. $$ ...
0
votes
1answer
3k views

Convert Recursive to Closed Formula

I got a particular sequence defined by the following recursive function: $$T_n = T_{n-1} \times 2 - T_{n-10}$$ I need help converting it to a closed form so I can calculate very large values of n ...
1
vote
4answers
121 views

Computing $\sum_{i=1}^{n}i^{k}(n+1-i)$

I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.
1
vote
1answer
168 views

Formula for $\sum_{k=1}^{n}{k^p}$ where p is a positive integer [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ Any hints that can take me from here or am I completely lost. ...
2
votes
1answer
219 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
4
votes
2answers
1k views

Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
25
votes
2answers
656 views

Computing $ \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$

I would like to compute: $$ \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{(-1)^{i+j}}{i+j}$$ I wanted to use Fubini's theorem for double series but $$ ...
2
votes
1answer
58 views

Finding an equivalent of $ u_{n}=\prod_{k=1}^{n} k^k $

I would like to find an equivalent of: $$ u_{n}=\prod_{k=1}^{n} k^k $$ I managed to find and asymptotic expansion of $ \ln(u_{n}) $ whose precision is $ o(n) $: $$ ...
2
votes
1answer
190 views

Number of combinations of $k$ numbers using arithmetic operations

What is the maximum number of positive rational values that can be obtained by combining $k$ positive integers using only addition, subtraction, multiplication, division, and parentheses? Assume that ...
37
votes
18answers
10k views

Getting the sequence $\{1, 0, -1, 0, 1, 0, -1, 0, \ldots\}$ without trig?

What's the simplest way to write a function that outputs the sequence: {1, 0, -1, 0, 1, 0, -1, 0, ...} ... without using any trig functions? I was able to come ...
1
vote
1answer
123 views

Domain of convergence of Mulitvariable series

What is region of convergence $(D\subset \mathbb C^2)$ of $$\sum_{n=0}^\infty(z_1^kz_2^l)^n$$ for fixed $k$ and $l$ integers. $z_1$ and $z_2$ are elements of complex plane. What is method of ...
10
votes
2answers
1k views

Convergence/divergence of $\sum\frac{a_n}{1+na_n}$ when $\sum a_n$ diverges.

A question from Rudin (Principles) Chapter 3. Let $a_n\geq0$ and $\sum a_n$ diverges. What can be said about convergence/divergence of $\sum\frac{a_n}{1+na_n}$? This one is being recalcitrant. Given ...
0
votes
3answers
541 views

Recursive sequence - Convergency and Limit

i just started university so im pretty new to all this new math. My problem is to solve this recursive sequence: $a_{n+1} = a_{n}^3$ with: $a_{0} = \frac{1}{2}$ ...
0
votes
2answers
107 views

How to show the sum of $\frac1{n^r}$ exists for all $r > 1$? [duplicate]

Possible Duplicate: Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$ How would I go about showing that $$\sum_{n=1}^{\infty } \frac{1}{n^{r}}$$ ...
0
votes
1answer
329 views

Set with $k$-subset unique sum

Probably, I should call it a sequence, anyway, is there a sequence/set (Fibonacci is a valid answer for this question (minus the first three Fibonacci numbers including zero), but too big) where any ...
3
votes
1answer
197 views

Compute floor sum

Write as a single sum: Given $\{a_n\}_n$, $a_i \in \mathbb{Q}$, $0 \lt a_i \le a_{i+1}$ $\sum_{i=1}^{n} \sum_{j=i+1}^{n} \lfloor a_j - a_i \rfloor$ I am not sure if this is possible. I know that ...
4
votes
1answer
234 views

Complex analysis: Coefficients of Laurent series

I have some past exam questions that I am confused with http://i39.tinypic.com/vuwxl.png sorry, can't embed images yet I'm not sure how to approach this, I'm completely lost and just attempted to ...
3
votes
2answers
146 views

Show the following series converges.

Let the real sequence ${x_n}$ be given by, $$\sum_{j=1}^{2n} \frac {1}{j} - \sum_{j=1}^{n} \frac {1}{j}. $$ Show that $0<x_{n}<x_{n+1}$ and that $x_{n}<1$ for all $n$. Deduce that $x_{n}$ ...
1
vote
1answer
54 views

Finding an equivalent of $u_{n}-u_{\infty} $ where $u_{n}= \sum_{k=1}^{n} \frac{n}{n^2+k^2} $

I would like to find an equivalent of $$ u_{n}-u_{\infty}=\sum_{k=1}^{n} \frac{n}{n^2+k^2}-u_{\infty} $$ Using Riemann sums, it is easy to show that: $$ u_{n} \sim \frac{\pi}{4}=u_{\infty} $$ ...
2
votes
1answer
157 views

About $\sum_{k=1}^\infty \frac{b_k}{k}$, where $b_k$ are Fourier coefficients

This is my first post here. I have some troubles with this property of the Fourier coefficients. Indeed, let $f(x)$ be a continuous real function, with compact support $[a,b] \subset (0,2\pi)$, and ...
9
votes
3answers
445 views

Series problem! Can someone give me a counterexample?

Suppose positive series $\sum a_n<+\infty$, does this implies that $$\lim_{n\to\infty}na_n=0 .$$
41
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
2
votes
1answer
105 views

Infinite Series with complex denominator

There is one series and it seems pretty much easy to check either it is divergent or convergent but because of the complex denominator I am not able to get the solution by the certain convergent ...
2
votes
2answers
149 views

Finding a closed formula for some SG function

Consider the following sequence, defined by recursion: $g(0)=g(1)=0$. If $n>1$, let $g(n)$ be the mex of the $g(k)$ with $\frac{n}{2} \leq k < n$. The first values of $g(n)$ with $0 \leq n ...
16
votes
1answer
597 views

Series which are not Fourier Series

How to show that $$ \sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n} $$ not the Fourier series of any function? I have shown that the series is convergent by Dirichlet test. Let $a(n)=\frac{1}{\log ...