Recurrence relations, convergence tests, identifying sequences
3
votes
2answers
122 views
How to simplify $f(x)=\sum\limits_{i=0}^{\infty}\frac{x^{i \;\bmod (k-1)}}{i!}$?
$$f(x)=\sum_{i=0}^{\infty}\frac{x^{i \;\bmod (k-1)}}{i!}$$
${i \bmod (k-1)}$ $\quad$ says the $x$ powers can be only $x^0$, $x^1$, ...,$x^{k-2}$
Understand simplify a way to transform this infinity ...
3
votes
2answers
472 views
Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$
How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$?
I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me.
(This series arises in ...
3
votes
2answers
413 views
Limit of a particular variety of infinite product/series
I was musing about a particular limit,
$L = \prod\limits_{n > 0} \bigl(1 - 2^{-n}\bigr)$:
we may bound 0.288 < L < 0.308, which we may show by taking the logarithm:
...
2
votes
1answer
153 views
Convergence of: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
2
votes
1answer
253 views
Evaluating 'combinatorial' sum
Help me please to calculate the following sum. I have seen such kind of formulas in the papers related to combinatorics, specifically 'trees'. I am curious how to calculate or approximate this sum:
...
2
votes
2answers
360 views
Equivalence of Completeness Axioms of Real Numbers
There are many equivalent versions of completeness in the real number system:
i) LUB/supremum property
ii) Monotone Convergence property
iii) Nested Interval property
iv) Bolzano Weierstrass property
...
1
vote
2answers
354 views
$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $[\epsilon, 2\pi - \epsilon]$
$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$, when ${n \to \infty}$. I need to prove that for every $\epsilon >0$, the series is uniformly converges within ...
1
vote
4answers
177 views
the sum of powers of $2$ between $2^0$ and $2^n$
Lately, I was wondering if there exists a closed expression for $2^0+2^1+\cdots+2^n$ for any $n$?
1
vote
4answers
271 views
Why is sum of a sequence $\displaystyle s_n = \frac{n}{2}(a_1+a_n)$?
Is there a way to prove that the sum of the arithmetic progression $a_1, a_2, \dots, a_n$ can be calculated by $\displaystyle s_n = \frac{n}{2}(a_1+a_n)$?
0
votes
2answers
113 views
What are common methods/techniques can be used to prove that limit of an infinite sequence exists?
I would like to know what are common methods can be used to show that an infinite sequence converges. From what I know so far,
If a sequence is bounded and monotonic increasing/decreasing then it ...
0
votes
6answers
255 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
205 views
Calculating the limit $\lim\limits_{k\to\infty} \prod\limits_{i=1}^{k}(1-\alpha_i+\alpha_i^2)$.
How do I evaluate $\displaystyle\lim_{k\to\infty} \prod_{i=1}^{k}(1-\alpha_i+\alpha_i^2)$?
Here, $\alpha_k\in (0,1)$ for every $k\in\mathbb{N}$ and $\displaystyle\lim_{k\to\infty}\alpha_k=0$.
0
votes
0answers
115 views
Stern's sequence [duplicate]
Possible Duplicate:
Summing the cubes of the insertion sequence
Could anyone help me determine the sum of cube of elements of the Stern's sequence?
Here is some information about it: ...
40
votes
1answer
1k views
Does $|n^2 \cos n|$ diverge to $+\infty$?
I was recently exposed to the problem of deciding whether
$$ \lim_{n \to +\infty} |n \cos n| = +\infty$$
where the limit is taken over the integers. As $|\cos n|$ oscillates throughout the interval ...
29
votes
2answers
3k views
Are there any series whose convergence is unknown?
Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
21
votes
1answer
633 views
How to prove convergence of polynomials in $e$ (Euler's number)
These polynomials in $e$ converge to 2$$f(i)=e^i - i \sum_{k=1}^{i-1}\frac{(i-k)^{k-1}{e^{i-k}}{(-1)^{k+1}}}{k!}, \text{ where } i>1$$
This function goes to 2. I've calculated this with sage math ...
21
votes
9answers
2k views
Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
Prove that for all integers $n$, $n \geq 1$,
$$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$
How would I go about proving this?
9
votes
2answers
397 views
How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$
How can I evaluate the following series:
$$1+\frac{3}{4}+\frac{3\cdot 5}{4\cdot 8}+\frac{3\cdot 5\cdot 7}{4\cdot 8\cdot 12}+\frac{3\cdot 5\cdot 7\cdot 9}{4\cdot 8\cdot 12\cdot 16}+\cdots$$
In one ...
10
votes
0answers
172 views
When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty ...
20
votes
2answers
1k views
Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof
I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$
$$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$
and thus (probably) ...
11
votes
1answer
289 views
Compute $\sum_{k=1}^{\infty}e^{-\pi k^2}\left(\pi k^2-\frac{1}{4}\right)$
How may I evaluate the below series?
$$\sum_{k=1}^{\infty}e^{-\pi k^2}\left(\pi k^2-\frac{1}{4}\right)$$
I'm supposed to come up with a solution by only using high school knowledge.
Thanks in advance ...
10
votes
2answers
392 views
Sum of the squares of the reciprocals of the fixed points of the tangent function
The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$.
I've seen this proved by means of residues, but I don't remember the details.
I've also ...
7
votes
1answer
94 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
14
votes
4answers
364 views
About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$
How to prove that
$$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$
$H_n$ is the n th harmonic number
13
votes
3answers
219 views
Proving a trig infinite sum using integration
How can I prove the following using integration and elementary functions?
Prove that:
$$\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \frac{\pi}{2} - \frac{\theta}{2}$$
$0 < \theta < 2\pi$
12
votes
4answers
809 views
Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges.
One can then ask the question of ...
9
votes
2answers
224 views
How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?
It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.
9
votes
4answers
404 views
How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
I can show that the following limit exists but
I am having difficulties to find it. It is
$$\lim_{n\to \infty} \sum_{k=1}^n \frac{k^n}{n^n}$$
Can someone please help me?
9
votes
2answers
672 views
Does $\sum \limits_{n=1}^\infty\frac{\sin n}{n}(1+\frac{1}{2}+\cdots+\frac{1}{n})$ converge (absolutely)?
I've had no luck with this one. None of the convergence tests pop into mind.
I tried looking at it in this form $\sum \sin n\frac{1+\frac{1}{2}+\cdots+\frac{1}{n}}{n}$ and apply Dirichlets test. I ...
17
votes
3answers
499 views
How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$?
This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this.
...
14
votes
2answers
805 views
De Moivre's Theorem. Motivation and origins.
I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous
$$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$
but ...
14
votes
2answers
690 views
Finding $f$ such that $ \int f = \sum f$
Please see the problem 5 of the given link: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=59&year=2005&sid=722231ab4ec5ce280584eb8f24f07656
It asks us to prove that ...
13
votes
4answers
802 views
Evaluating $ \lim\limits_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $
How would you evaluate the following series?
$$\lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $$
Thanks.
11
votes
2answers
188 views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
10
votes
1answer
911 views
Uniform convergence of series $\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$
Using Dirichlet series test I've proves that series $\displaystyle\sum\limits_{n=2}^\infty\frac{\sin n x}{n\log n}$ converges for all $x\in\mathbb{R}$.
How to determine whether the series ...
10
votes
3answers
258 views
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
Let $\sum a_{n}$ be an absolutely convergent series such that $$\sum a_{kn}=0$$ for all $k\geq 1$. Help me prove that $a_{n}=0$ for all $n$.
Thank you!
7
votes
5answers
266 views
Finding sum of a series: difference of cubes
I am trying to find sum of the infinite series:
$$1^{3}-2^{3}+3^{3}-4^{3}+5^{3}-6^{3} + \ldots$$
I tried to solve it by subtracting sum of even cubes from odd, but that solves only half of the ...
6
votes
3answers
622 views
Explain why calculating this series could cause paradox?
$$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + ...
5
votes
2answers
457 views
Limit of $S(n) = \sum_{k=1}^{\infty} \left(1 - \prod_{j=1}^{n-1}\left(1-\frac{j}{2^k}\right)\right)$ - Part II
This is a follow up of Limit of $S(n) = \sum_{k=1}^{\infty} \left(1 - \prod_{j=1}^{n-1}\left(1-\frac{j}{2^k}\right)\right)$
More details can be found in the above thread.
Let $S(n) = \displaystyle ...
14
votes
2answers
606 views
Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$
I'm interested in finding an elementary proof for the following sum inequality:
$$\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$$
If this inequality is easy to prove, then one may easily prove that the sum ...
13
votes
3answers
1k views
Non-increasing sequence of positive real numbers with prime index
If $a_n$ is a sequence of non-increasing positive numbers, then suppose we already know that
$$\sum_p a_p$$ converges, when $p$ runs over the primes, what should be used to prove that $$\sum_n ...
13
votes
3answers
400 views
challenging alternating infinite series involving $\ln$
I ran across an infinite series that is allegedly from a Chinese math contest.
Evaluate:
$\displaystyle\sum_{n=2}^{\infty}(-1)^{n}\ln\left(1-\frac{1}{n(n-1)}\right).$
I thought perhaps this ...
12
votes
2answers
1k views
Is there is any way to calculate generic formula for $1! +2! +3! + \cdots + n! $?
I came across a question where I need to find the sum of factorial of the first $n$ numbers.
So I was wondering if there is any generic formula for this?
Like there is a generic formula for different ...
10
votes
1answer
165 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
6
votes
4answers
318 views
Convergence of $\sum_{k=1}^{n} f(k) - \int_{1}^{n} f(x) dx$
I had asked this question sometime ago here: http://math.stackexchange.com/questions/2536/finding-f-such-that-int-f-sum-f
Now i have a question which i think is more or less related to it.
Let $f$ ...
4
votes
6answers
557 views
Fastest Square Root Algorithm
What is the fastest algorithm for finding the square root of a number?
I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
4
votes
1answer
150 views
Infinite series representation. Limited or not?
Recently I've found the following.
Let $n < m$. Then the integral
$$\int\limits_0^\infty {\frac{{{x^{n-1}}}}{{1 + {x^m}}}} dx$$
converges and its value is (using the $B$ and $\Gamma$ function ...
9
votes
1answer
180 views
Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$
Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$
converges?
I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
8
votes
3answers
192 views
Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$
Evaluate the sum:
$$\sum_{n=0}^{\infty} \frac{1}{F_{(2^n)}}$$
where $F_{m}$ is the $m$-th term of the Fibonacci sequence. I need some support here. Thanks.
8
votes
2answers
296 views
Sum of the reciprocal of sine squared
I encountered an interesting identity when doing physics homework, that is,
$$ \sum_{n=1}^{N-1} \frac{1}{\sin^2 \dfrac{\pi n}{N} } = \frac{N^2-1}{3}. $$
How is this identity derived? Are there any ...
