Recurrence relations, convergence tests, identifying sequences
131
votes
17answers
8k views
Different methods to compute $\sum\limits_{n=1}^\infty \frac{1}{n^2}$
As I have heard people did not trust Euler when he first discovered the formula
$$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$
However, Euler was Euler and he gave other proofs.
I ...
60
votes
6answers
1k views
Contest problem about convergent series
The following is probably a math contest problem. I have been unable to locate the original source.
Suppose that $\{a_i\}$ is a set of positive real numbers and the series $$\sum_{n = 1}^\infty ...
60
votes
6answers
3k views
Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing
Prove without calculus that the sequence
$$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$
is strictly decreasing.
56
votes
8answers
2k views
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
To prove the convergence of
$$\sum_{n=1}^{\infty} \frac1{n^p}$$
for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.
I am wondering if there is a ...
48
votes
7answers
4k views
What's next in this number series?
340, 680, 1428, 3141.6, _____
This is from an aptitude test. I'm not able to find any pattern in them.
48
votes
5answers
3k views
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.
If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
46
votes
17answers
2k views
Proving the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ without induction
I recently proved that
$$
\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2
$$
Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
41
votes
3answers
616 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
...
40
votes
5answers
6k views
Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
...
40
votes
1answer
1k views
Does $|n^2 \cos n|$ diverge to $+\infty$?
I was recently exposed to the problem of deciding whether
$$ \lim_{n \to +\infty} |n \cos n| = +\infty$$
where the limit is taken over the integers. As $|\cos n|$ oscillates throughout the interval ...
40
votes
1answer
1k views
What's the value of this Vieta-style product involving the golden ratio?
One way of looking at the Vieta product
$${2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$
is as the infinite product of a series of successive ...
39
votes
4answers
2k views
Double sum - Miklos Schweitzer 2010
There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is:
Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 ...
38
votes
7answers
4k views
Infinity = -1 paradox
I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:
Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 ...
36
votes
7answers
783 views
Problems regarding $\{x_n \}$ defined by $x_1=1$; $x_n$ is the smallest distinct natural number such that $x_1+…+x_n$ is divisible by $n$.
Let me denote a sequence of distinct natural numbers by $x_n$ whose terms are determined as follows:
$x_1$ is $1$ and $x_2$ is the smallest distinct natural number $n$ such that $x_1+x_2$ is divisible ...
35
votes
19answers
5k views
Getting the sequence $1, 0, -1, 0, 1, 0, -1, 0, \ldots$ without trig?
What's the simplest way to write a function that outputs the sequence:
1, 0, -1, 0, 1, 0, -1, 0, ...
... without using any trig functions?
I was able to come up ...
35
votes
4answers
3k views
Value of $\sum\limits_n x^n$
Why is $\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n$ equal $1/(1-0.7) = 10/3$ ?
Can we generalize the above to
$\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?
Are there some ...
35
votes
6answers
1k views
Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge?
Can someone give a simple explanation for why the harmonic series
$$\frac 1 1 + \frac 12 + \frac 13 + \cdots $$
doesn't converge, but just grows very slowly?
I'd prefer an easily ...
35
votes
4answers
2k views
“Closed” form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating ...
34
votes
5answers
1k views
Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$
$$ \sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$$
Does it converge or diverge? Can we have a rigorous proof that is not probabilistic? For reference, this question is supposedly a mix of real ...
33
votes
10answers
1k views
What it the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?
What it the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$?
Using the definition:
$$\lim_{n\rightarrow\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$
I finally get 2 ...
33
votes
2answers
1k views
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
Consider the sequence defined as
$x_1 = 1$
$x_{n+1} = \sin x_n$
I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't ...
33
votes
2answers
708 views
Is there a function with the property $f(n)=f^{(n)}(0)$?
Is there a not identically zero, real-analytic function $f:\mathbb{R}\rightarrow\mathbb{R}$, which satisfies
$$f(n)=f^{(n)}(0),\quad n\in\mathbb{N} \text{ or } \mathbb N^+?$$
What I got so far:
Set
...
32
votes
6answers
1k views
A question on Taylor Series and polynomial
Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$.
For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial.
Is true that $f(x)$ is a ...
31
votes
5answers
899 views
How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?
How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$
I found the above interesting identity in the book $\bf \pi$ Unleashed.
Does anyone knows how to ...
31
votes
1answer
374 views
Is this sequence unbounded?
Problem.
Suppose that a sequence $\{a_n\}_{(n\ge 1)}$ is a strict increasing sequence of positive integers such that
$$\forall i,\phantom{;}j(i\neq j);\phantom{;}a_i \not\mid a_j$$
Prove that ...
30
votes
9answers
2k views
Prove that 16, 1156, 111556, 11115556, 1111155556… are squares.
I'm 16 years old, and I'm studying for my exam maths coming this monday. In the chapter "sequences and series", there is this exercise:
Prove that a positive integer formed by $k$ times digit 1, ...
30
votes
4answers
2k views
The 9 Billion Names of God
TLDR; I go on a math adventure and get overwhelmed :)
Some background:
My maths isn't great (I can't read notation) but I'm a competent programmer and reasonable problem solver. I've done the first ...
30
votes
3answers
813 views
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration.
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}.
$$
I can see that:
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
30
votes
4answers
542 views
Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$
It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
30
votes
1answer
466 views
Prove that $\sum_{k=1}^{\infty} \large\frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$
Prove that
$$\sum_{k=1}^{\infty} \frac{k}{\text{e}^{2\pi k}-1}=\frac{1}{24}-\frac{1}{8\pi}$$
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 ...
29
votes
1answer
601 views
In which ordered fields does absolute convergence imply convergence?
In the process of touching up some notes on infinite series, I came across the following "result":
Theorem: For an ordered field $(F,<)$, the following are equivalent:
(i) Every Cauchy ...
28
votes
5answers
2k views
Does $\zeta(3)$ have a connection with $\pi$?
The problem
Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)?
Details
Several ...
28
votes
5answers
873 views
$\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate:
$$\lim_{n\rightarrow\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$
By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. ...
28
votes
4answers
2k views
$n$th derivative of $e^{1/x}$
I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula
$$\frac{\mathrm d^n}{\mathrm ...
28
votes
2answers
559 views
Predicting Real Numbers
Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
25
votes
4answers
1k views
How to sum this series for $\pi/2$ directly?
The sum of the series
$$
\frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1}
$$
can be derived by accelerating the Gregory Series
$$
\frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2}
...
25
votes
4answers
3k views
Nice proofs of $\zeta(4) = \pi^4/90$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
25
votes
2answers
492 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 $$ ...
25
votes
3answers
371 views
Sequence of numbers with prime factorization $pq^2$
I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions:
What is the ...
24
votes
3answers
789 views
Tall fraction puzzle
I was given this problem 30 years ago by a coworker, posted it 15 years ago to rec.puzzles, and got a solution from Barry Wolk, but have never seen it again. Consider the series: $$1, ...
24
votes
1answer
690 views
A question about series with a strange property.
Does there exist a sequence $\left(a_n\right)_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}, \forall n=1,2,3,\ldots$ such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges?
This is the ...
24
votes
3answers
1k views
Asymptotic (divergent) series
MOTIVATION. After having read in detail an article by Alf van der Poorten I read a very short paper by Roger Apéry. I am interested in finding a proof of a series expansion in the latter, which is in ...
23
votes
3answers
978 views
Why is this series of square root of twos equal $\pi$?
Wikipedia claims this but only cites an offline proof:
$$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$
for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
23
votes
2answers
469 views
Rounding is asymptotically useless?
Recently I came across the nice result that
$$\left\lfloor n \right\rfloor - \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor - \left\lfloor \frac{n}{4}\right\rfloor + ...
23
votes
4answers
507 views
If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?
Let $a_n$ be a sequence of complex numbers that converge to zero. Can we always find $s_n \in \{-1,1\}$ such that $\sum_{n=1}^{\infty} s_n a_n$ converges?
If the $a_n$ are real numbers, we can find ...
23
votes
2answers
631 views
On calculating $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ — where is the mistake?
I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...
23
votes
2answers
974 views
Why do some Fibonacci numbers appear in an approximation for $e^{\pi\sqrt{163}}$?
It is rather well-known that,
$e^{\pi\sqrt{43}} \approx 960^3 + 743.999\ldots$
$e^{\pi\sqrt{67}} \approx 5280^3 + 743.99999\ldots$
$e^{\pi\sqrt{163}} \approx 640320^3 + 743.999999999999\ldots$
Not ...
22
votes
15answers
3k views
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really ...
22
votes
3answers
566 views
Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums.
...
