Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
65,955
questions
860
votes
54
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Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler ...
452
votes
24
answers
91k
views
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
- 4,735
446
votes
10
answers
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My son's Sum of Some is beautiful! But what is the proof or explanation?
My youngest son is in $6$th grade. He likes to play with numbers. Today, he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, ...
- 3,441
268
votes
9
answers
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Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate:
$$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$
By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$, which is a pretty interesting and nice ...
- 44.4k
246
votes
12
answers
48k
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 $\frac 1p$ ...
- 4,640
235
votes
6
answers
84k
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When can you switch the order of limits?
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \...
- 9,822
198
votes
8
answers
38k
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 ...
- 2,271
189
votes
10
answers
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Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
To prove the convergence of the p-series
$$\sum_{n=1}^{\infty} \frac1{n^p}$$
for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.
I am wondering if ...
- 2,824
188
votes
28
answers
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Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ 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 interpretation ...
- 5,897
183
votes
0
answers
5k
views
Sorting of prime gaps
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$
If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new ...
- 10.3k
177
votes
7
answers
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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:
$$f(x)=\...
- 1,915
166
votes
26
answers
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Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
Can someone give a simple explanation as to why the harmonic series
$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$
doesn't converge, on the other hand it grows very slowly?...
- 9,805
160
votes
4
answers
10k
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Sum of random decreasing numbers between 0 and 1: does it converge??
Let's define a sequence of numbers between 0 and 1. The first term, $r_1$ will be chosen uniformly randomly from $(0, 1)$, but now we iterate this process choosing $r_2$ from $(0, r_1)$, and so on, so ...
- 2,597
154
votes
14
answers
21k
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Why does an argument similiar to 0.999...=1 show 999...=-1?
I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc.
Can anyone point me to resources that would explain what the ...
- 1,605
153
votes
33
answers
88k
views
Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really ...
- 1,883
144
votes
16
answers
11k
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Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums.
...
- 55.8k
144
votes
1
answer
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Identification of a curious function
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},
$$
where $...
- 57.3k
141
votes
36
answers
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$
Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
- 1,585
135
votes
7
answers
108k
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Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=...
123
votes
18
answers
53k
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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
- 55.8k
122
votes
4
answers
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views
Motivation for Ramanujan's mysterious $\pi$ formula
The following formula for $\pi$ was discovered by Ramanujan:
$$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$
Does anyone know how it works, or ...
- 19.1k
117
votes
1
answer
4k
views
Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$
Is there a way to assess the convergence of the following series?
$$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$
From numerical estimations it seems to be convergent but I don't know how to prove it.
- 1,859
115
votes
25
answers
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Can an infinite sum of irrational numbers be rational?
Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear ...
- 1,841
115
votes
5
answers
8k
views
How to sum this series for $\pi/2$ directly?
The sum of the series
$$
\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}
...
- 348k
113
votes
16
answers
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If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum_{n=1}^\infty a_n<\infty$, then $\lim\limits_{n\to\infty}{n a_n} = 0$
I'm studying for qualifying exams and ran into this problem.
Show that if $\{a_n\}$ is a nonincreasing sequence of positive real
numbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \...
- 4,666
110
votes
6
answers
20k
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Why is a geometric progression called so? [duplicate]
Just curious about why geometric progression is called so. Is it related to geometry?
- 1,401
108
votes
3
answers
4k
views
How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$
Nice problem:
Let $a_{0}=1$ and
$$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that
$$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\...
- 93.6k
104
votes
13
answers
9k
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Limit of sequence in which each term is defined by the average of preceding two terms
We have a sequence of numbers $x_n$ determined by the equality
$$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$
The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $...
- 2,277
103
votes
4
answers
5k
views
A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
As the title says, I would like to launch a community project for
proving that the series $$\sum_{n\geq 1}\frac{\sin(2^n)}{n}$$ is
convergent.
An extensive list of considerations follows. The ...
- 355k
101
votes
1
answer
4k
views
Arithmetic-geometric mean of 3 numbers
The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows:
$$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,...
- 47.3k
100
votes
14
answers
54k
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How to find a general sum formula for the series: 5+55+555+5555+.....?
I have a question about finding the sum formula of n-th terms.
Here's the series:
$5+55+555+5555$+......
What is the general formula to find the sum of n-th terms?
My attempts:
I think I need to ...
- 2,551
99
votes
6
answers
9k
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Can the product of infinitely many elements from $\mathbb Q$ be irrational?
I know there are infinite sums of rational values, which are irrational (for example the Basel Problem). But I was wondering, whether the product of infinitely many rational numbers can be irrational. ...
- 1,257
97
votes
2
answers
34k
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Predicting Real Numbers
Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
97
votes
2
answers
2k
views
When does a sequence of rotated-and-circumscribed rectangles converge to a square?
Recently I came up with an algebra problem with a nice geometric representation. Basically, I would like to know what happens if we repeatedly circumscribe a rectangle by another rectangle which is ...
- 773
95
votes
2
answers
6k
views
Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing
Prove without calculus that the sequence
$$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$
is strictly decreasing.
- 44.4k
93
votes
6
answers
3k
views
Contest problem: Show $\sum_{n = 1}^\infty \frac{n^2a_n}{(a_1+\cdots+a_n)^2}<\infty$ s.t., $a_i>0$, $\sum_{n = 1}^\infty \frac{1}{a_n}<\infty$ [closed]
The following is probably a math contest problem. I have been unable to locate the original source.
Suppose that $\{a_i\}$ is a sequence of positive real numbers and the series $\displaystyle\sum_{n ...
- 40.5k
92
votes
4
answers
6k
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How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?
How can one prove this identity?
$$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$
There is a formula for $\zeta$ values at even integers, but it ...
- 23.4k
90
votes
8
answers
19k
views
Is there a size of rectangle that retains its ratio when it's folded in half?
A hypothetical (and maybe practical) question has been nagging at me.
If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches ...
- 1,011
89
votes
4
answers
46k
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Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
Is the following true?
Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a
further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
I ...
- 933
88
votes
9
answers
10k
views
Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$
I found the following formula
$$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$
and it is cited that Euler proved the ...
- 14.4k
86
votes
5
answers
24k
views
Limit of the nested radical $x_{n+1} = \sqrt{c+x_n}$
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)
For $c \gt 0$, consider the quadratic equation
$x^2 - x - c = 0, x > 0$.
Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, ...
- 1,005
86
votes
1
answer
2k
views
Conjectured formula for the Fabius function
The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions:
a functional–integral equation$\require{action}
\require{enclose}{^{\texttip{\dagger}{a ...
- 47.3k
85
votes
0
answers
2k
views
Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?
The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
- 55.1k
84
votes
3
answers
11k
views
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
Consider the sequence defined as
$x_1 = 1$
$x_{n+1} = \sin x_n$
I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't ...
- 82.6k
82
votes
3
answers
4k
views
How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$
Question:
Show that $$A=\lim_{n\to \infty}\sqrt{1+\sqrt{\dfrac{1}{2}+\sqrt{\dfrac{1}{3}+\cdots+\sqrt{\dfrac{1}{n}}}}}$$
exists, and find the best estimate limit $A$.
It is easy to show that
...
- 93.6k
81
votes
13
answers
8k
views
What is an example of a sequence which "thins out" and is finite?
When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes. Usually, they start getting reeled in ...
- 1,662
81
votes
2
answers
3k
views
Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$:
Proof of equality ...
- 16k
80
votes
12
answers
11k
views
The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
What is the sum of the 'second half' of the harmonic series?
$$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$
More precisely, what is the limit of the above sequence of partial sums?
- 4,811
80
votes
8
answers
11k
views
How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?
How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$
I found the above interesting identity in the book $\bf \pi$ Unleashed.
Does anyone knows how to ...
- 5,627
78
votes
6
answers
13k
views
Is there a slowest rate of divergence of a series?
$$f(n)=\sum_{i=1}^n\frac{1}{i}$$
diverges slower than
$$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$
, by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $...
- 6,413