Recurrence relations, convergence tests, identifying sequences
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 ...
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 ...
17
votes
8answers
1k views
How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$
How can if find the sum for:
$$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$
I know the answer thanks to Wolfram Alpha. I'm more concerned with how to get to to that number. It cites tests to prove ...
15
votes
19answers
7k views
Proof for formula for sum of sequence $1+2+3+\ldots+n$?
Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.
How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? ...
14
votes
2answers
727 views
Possibility to simplify $\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{\pi }{{\sin \pi a}}} $
Is there any way to show that
$$\sum\limits_{k = - \infty }^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{a + k}} = \frac{1}{a} + \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}\left( ...
35
votes
4answers
2k views
“Closed” form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating ...
20
votes
8answers
926 views
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$?
I know this is well-known. But how to prove it rigorously? Even mathematical induction does not seem so straight-forward.
...
46
votes
17answers
2k views
Proving the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ without induction
I recently proved that
$$
\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2
$$
Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
48
votes
5answers
3k views
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.
If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
4
votes
1answer
747 views
Finite Sum of Power?
Sorry for the remedial math but:
Can someone tell me how to get a closed form for
$$\sum_{k=1}^n k^x$$
For $x = 1$, it's just the classic $n(n+1)/2$. What is it for $x > 1$?
22
votes
15answers
3k views
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really ...
6
votes
2answers
809 views
$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)
For $c \gt 0$, consider the quadratic equation
$x^2 - x - c = 0, x > 0$.
Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and ...
56
votes
8answers
2k views
Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$
To prove the convergence of
$$\sum_{n=1}^{\infty} \frac1{n^p}$$
for $p > 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.
I am wondering if there is a ...
33
votes
2answers
1k views
Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$
Consider the sequence defined as
$x_1 = 1$
$x_{n+1} = \sin x_n$
I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't ...
7
votes
3answers
623 views
Find the sum of $\sum 1/(k^2 - a^2)$ when $0<a<1$
So I have been trying for a few days to figure out the sum of
$$ S = \sum_{k=1}^\infty \frac{1}{k^2 - a^2} $$ where $a \in (0,1)$. So far from my nummerical
analysis and CAS that this sum equals
$$ ...
17
votes
7answers
995 views
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$
I.e - does the second half of the harmonic series go to zero?
I know that for a finite number of terms the limit of the sum is ...
5
votes
2answers
243 views
Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials
Suppose that $$Q(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} $$and $$P(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}.$$ How do I find $$\lim_{x\rightarrow\infty}\frac{Q(x)}{P(x)}$$ and what ...
15
votes
11answers
2k views
$ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
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 ...
17
votes
1answer
1k views
Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
Here is an exercise, on analysis which i am stuck.
How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on ...
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. ...
13
votes
4answers
1k views
Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$ [duplicate]
Possible Duplicate:
On the sequence $x_{n+1} = \sqrt{c+x_n}$
Where does this sequence converge?
$\sqrt{7},\sqrt{7+\sqrt{7}},\sqrt{7+\sqrt{7+\sqrt{7}}}$,...
5
votes
3answers
388 views
Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$
How to show the following equality?
$$\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$$
3
votes
8answers
410 views
How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?
Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$
how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$
Thank you
22
votes
3answers
563 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.
...
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 ...
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 ...
13
votes
4answers
712 views
Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$
So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
5
votes
1answer
268 views
Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means
Prove that if $\lim_{n \to \infty}z_{n}=A$ then:
$$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$
I was thinking spliting it in: ...
15
votes
4answers
4k views
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 ...
8
votes
3answers
334 views
A log improper integral
Evaluate :
$$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$
I found it can be simplified to
$$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$
I found the exact value in the ...
16
votes
3answers
660 views
Result of the product $0.9 \times 0.99 \times 0.999 \times …$
My question has two parts:
How can I nicely define the infinite sequence $0.9,\ 0.99,\ 0.999,\ ...$? One option would be this recursive definition below; is there a nicer way of doing this? Maybe ...
7
votes
3answers
3k views
Predict next number from a series
Which methods I can use to predict next number from a series of numbers ?
I know the min & max possible number in advance.
20
votes
3answers
738 views
Generalizing $\sum \limits_{n=1}^{\infty }n^{2}/x^{n}$ to $\sum \limits_{n=1}^{\infty }n^{p}/x^{n}$
For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate
...
7
votes
5answers
180 views
Given $y_n=(1+\frac{1}{n})^{n+1},n \in \mathbb{N},n \geq 1$ Show that $\lbrace y_n \rbrace$ is a decrasing sequence
Given
$$
y_n=\left(1+\frac{1}{n}\right)^{n+1}\hspace{-6mm},\qquad n \in \mathbb{N}, \quad n \geq 1.
$$
Show that $\lbrace y_n \rbrace$ is a decrasing sequence. Anyone can help ? I consider the ratio ...
5
votes
4answers
173 views
The generating function for the Fibonacci numbers
$$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$
The coefficients are Fibonacci numbers $\left\{1,1,3,5,8,13,21,...\right\}$.
Please HELP. Thanks guys.
2
votes
3answers
714 views
I have a problem understanding the proof of Rencontres numbers (Derangements)
I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation
$$D_{n,0}=\left[\frac{n!}{e}\right]$$
where $[\cdot]$ denotes the rounding function (i.e., ...
2
votes
1answer
110 views
Summation of natural number set with power of $m$
Who knows about the summation of this series:
$$\sum\limits_{i=1}^{n}i^m $$ where $m$ is constant and $m\in \mathbb{N}$?
thanks
21
votes
1answer
2k views
Convergence of $\sum \limits_{n=1}^{\infty}\sin(n^k)/n$
Does $S_k= \sum \limits_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$?
Motivation: I recently learned that $S_1$ converges. I think $S_2$ converges by the integral test. Was the question ...
14
votes
5answers
1k views
Crazy induction
I found crazy (for me at least) induction example, in fact it just would be nice to prove. (Even have problems with starting) Any hints are highly valued: ...
2
votes
1answer
555 views
Inequality between $\ell^p$-norms
Suppose that a sequence $x=(x_n)$ belongs both to $\ell^p$ and $\ell^q$ ($p,q>1$, $p\neq q$).
Is there any inequality between $\|x\|_p$ and $\|x\|_q$. Can one $\ell^p$ be continuously embedded into ...
15
votes
5answers
769 views
Find a closed form for this sequence: $a_{n+1} = a_n + a_n^{-1}$
Today, we had a math class, where we had to show, that $a_{100} > 14$ for
$$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$
Apart from this task, I asked my self: Is there a closed form for this ...
5
votes
3answers
479 views
Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$
How does one sum the series, $$ S = a -\frac{2}{3}a^{3} + \frac{2 \cdot 4}{3 \cdot 5} a^{5} - \frac{ 2 \cdot 4 \cdot 6}{ 3 \cdot 5 \cdot 7}a^{7} + \cdots $$
This was asked to me by a high school ...
7
votes
5answers
483 views
Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]
Possible Duplicate:
Summation of a factorial
This equation is given:
$$
1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1
$$
I've solved it using mathematical induction but ...
8
votes
3answers
277 views
Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
2
votes
2answers
391 views
limit of quotient of two series
Let $Y_{n} > 0$ for all $ n\in \mathbb{N} $, with $\sum{Y_{n}}= +\infty$.
If $\displaystyle\lim\limits_{n\rightarrow \infty}\frac{X_{n}}{Y_{n}}= a$ then ...
1
vote
2answers
169 views
prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:
$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$
$Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$
$p=\int_{-1}^{1}Z(x)dx$
...
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 ...
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 ...
30
votes
4answers
541 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 $n = 6$; ...
