Tagged Questions
0
votes
3answers
35 views
Arithmetic progression where the sum of the first $p$ terms is $q$, and the first $q$ terms is $p$
The sum of first $p$ terms of an arithmetic progression (A.P) is $q$, and the sum of the first $q$ terms of the same A.P. is $p$.
Find the sum of the first $p+q$ terms of the A.P.
0
votes
1answer
35 views
How to derive the sum of an arithmetic sequence?
I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the ...
2
votes
3answers
58 views
Series Summation
I have the series
$$\sum_{k=1}^{N-1}\frac{1}{1-w_k} $$ where $w_k=e^{\frac{2\pi i k}{N}}$, how can I find the summation of this , another question related to this sum
$$\sum_{k=1}^{N-1}\frac{1}{z-w_k} ...
1
vote
2answers
56 views
Partial fraction expansion two variables
How to expand
$$\frac{y}{(x-y)(y-1)}$$
by partial fraction expansion.
6
votes
2answers
74 views
Evaluate the Sum $\sum_{i=0}^\infty \frac {i^N} {4^i}$ [duplicate]
$$\displaystyle\sum_{i=0}^\infty \frac {i^N} {4^i}$$
I'm supposed to evaluate this as I'm working through Data Structures and Algorithm Analysis in C++. I've solved similar problems, and after ...
0
votes
0answers
13 views
Generalization/abstraction of an allocation problem
I'm having difficulty generalizing and finding the right abstraction for the following real world problem :
For each period, on a global segment, I know some data that is divided into intermediate ...
1
vote
2answers
58 views
Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
Consider the sequence $\{u_n\}$ where $u_0=1,u_1=1-a$ for some $0< a < 1/4$, and $u_{n+2} = u_{n+1}-au_n$. Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
1
vote
2answers
64 views
Series, Looks Simple, but I am Stuck
I promised a friend that I could help her about math questions. Yet, I am stuck with a series question. I have written the open form of each term. And I have split the general term into multiples. I ...
0
votes
2answers
31 views
Investigate monotony, bound and convergence
I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it.
The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
0
votes
0answers
38 views
prove sum property by induction
Knowing that $(a_i)_{i\ge1}$ prove that $\forall n \in \Bbb N$:
$$\sum^n_{i=1}ra_i=r\Big(\sum^n_{i=1}a_i \Big)$$
This kind of demonstrations is totally strange to me, I do not understand how to ...
1
vote
4answers
88 views
What is the formula of: $a^{0} + a^{1} + a^{2} + … + a^{n-1} + a^{n}$? [duplicate]
What is the formula of:
$$a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n}$$
Any ideas?
3
votes
5answers
172 views
Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum
Is there a formula for the following sum?
$S_n = 1\cdot2 + 2\cdot 3 + 3\cdot 4 + 4\cdot 5 +\ldots + n\cdot (n+1)$
1
vote
0answers
29 views
Strange equality of the operator E($Eu_n=u_{n+1}$)
The operator $E$ is defined as $Eu_n=u_{n+1}$.
I encountered a strange equality. when I tried out
Let $u_n$ represent a series such that
$$u_{n+2}=u_{n+1}+u_n. \tag{$\star$}$$
Or
...
1
vote
3answers
73 views
Find a formula that generates the following sequence
Find a formula which generates the following sequence.
$$15,20,25,30,35 \ldots $$
The answer is $5(n + 2) $
How? I know it comes from the formula $a_n = a_1 + (n - 1) d$, but I am not sure how they ...
0
votes
3answers
56 views
Arithmetic and Geometric Sequences (with already-known ratio/difference)
Arithmetic sequence: common difference = $10$
Geometric sequence: common ratio = $2$
Arithmetic:
$f(1) = 50$
$f(2) = 60$
$f(3) = 70$
$f(4) = 80$ ...
2
votes
4answers
129 views
Prove without induction : $\sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=1}^n \frac{1}{k+n}$
Prove without induction that : $$
\sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=1}^n \frac{1}{k+n}
$$
Please if you have any elementary tricks just post hints.
0
votes
0answers
41 views
logistic difference equation problem
Consider logistic difference equation
$${{x}_{n+1}}-r{{x}_{n}}\left( 1-{{x}_{n}} \right)=f\left( x \right),\ \ 0\le {{x}_{n}}\le 1\ \ \ \ \ \ \left( 1 \right)$$
1.Show hat expression $$f\left( f\left( ...
6
votes
5answers
115 views
Is the expression $\sum_{n=1}^{\infty}\prod_{k=1}^{n}\left(1-\frac{2}{3k}\right)$ bounded?
Today I have encounter an integral:
$$\int_0^{\infty}\left[\frac{1}{3}\frac{\sin x}{x}+\cdots+\frac{1\times4\times\cdots\times(3n-2)}{3^nn!}\left(\frac{\sin x}{x}\right)^n+\cdots\right]\text{d}x$$
...
3
votes
2answers
165 views
A limit about euler's constant
Show that :
$$\lim_{m\to \infty}\left[ -\frac{1}{2m}+\ln \left( \frac{\text{e}}{m} \right)+\sum\limits_{n=2}^m \left( \frac{1}{n}-\frac{\zeta \left( 1-n \right)}{m^n} \right) \right]=\gamma $$
How to ...
10
votes
0answers
93 views
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes.
I've been thinking about this question, but ...
0
votes
3answers
34 views
Series and Sequences
Is there any formula for the series below:
$$
(2^0)n + (2^1)(n-1) + (2^2)(n-2) + (2^3)(n-3) + \cdots +(2^{n-1})(1)
$$
If no, please let me know how to solve such series
1
vote
0answers
43 views
find all sequence satisfies $y^n_1 + 2y^n_2 + · · · + 2011y^n_{2011} = a^{n+1}+1$
Determine all sequences $(y_1, y_2, . . . , y_{2011})$ of positive integers such that for every positive integer $n$ there exist an integer a with $y^n_1 + 2y^n_2 + · · · + 2011y^n_{2011} = ...
0
votes
3answers
118 views
Showing that $X_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ is not bounded above. [duplicate]
Possible Duplicate:
Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge?
Prove that the sequence converges
I have to show that $X_n$ is not bounded above,
...
3
votes
2answers
77 views
3
votes
8answers
195 views
Is $X_n = 1+\frac12+\frac{1}{2^2}+\cdots+\frac{1}{2^n}; \forall n \ge 0$ bounded?
Is $X_n = 1+\frac12+\frac{1}{2^2}+\cdots+\frac{1}{2^n}; \forall n \ge 0$ bounded?
I have to find an upper bound for $X_n$ and i cant figure it out, a lower bound can be 0 or 1 but does it have an ...
2
votes
4answers
333 views
Find the number of digits, $D$, in the decimal expansion of the large number $N=4^{4^{4^{4}}}$
The full question is:
Find the number of digits, $D$, in the decimal expansion of the large
number
$$N=4^{4^{4^{4}}}$$
Try and find the most efficient ways of finding $D$.
I know that ...
6
votes
3answers
160 views
Help with infinite sum
Can you guys give me a hint on evaluating $$\sum_{n=1}^\infty \frac{1}{n(n+2)(n+4)}?$$
I have tried partial fractions but the series is not telescopic (at least I cannot see it)...
1
vote
2answers
73 views
Closed Form for Simple Looking Sum?
Is there a closed form for the sum below?
$$\sum_{s=0}^{m-1} \sum_{t=0}^{m-1} s~t~(m-s)~(m-t)\left|s-t\right|$$
0
votes
1answer
50 views
Converting loop to a closed form expression? [duplicate]
Possible Duplicate:
How to convert this loop into a closed form expression?
I have the following code in Python
...
2
votes
1answer
70 views
Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.
Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by
$$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$
Find $\lambda>1$ such that $a_n=0$.
The answer is ...
5
votes
1answer
47 views
Showing that a progression is arithmetic
this one is from Gelfand's book "Algebra".
Problem 204. Is it possible that numbers $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{5}$ are (not necessarily adjacent) terms of the same arithmetic ...
3
votes
2answers
156 views
Repeating Square Root Simplification
Alright, so I have a question on a little open-book challenge-test thingy that deals with repeating square roots, in a form as follows...
$\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$
Repeated 2012 times ...
-1
votes
2answers
51 views
How is $\frac{2x+6}{(x+2)^3}=\frac{2}{(x+2)^2}+\frac{2}{(x+2)^3}$
$$\begin{align}
|R_5|&\leq \int_5^\infty\frac{2x+6}{(x+2)^3}dx\\
&=\int_5^\infty\left(\frac{2}{(x+2)^2}+\frac{2}{(x+2)^3}\right)dx\text{(How did we get to this step?)}\\
...
1
vote
1answer
66 views
Partial Fractions continued…
Hi asked the following question yesterday: Obtaining the sum of a series
Given the answers to that question by wj32, I am now trying to solve the following problem:
Consider the series
...
0
votes
2answers
52 views
1
vote
0answers
65 views
Simplifying this infinite series [duplicate]
Possible Duplicate:
How can I evaluate $\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$
I have an infinite series like so:
$$\sum_{i=0}^\infty (i+1)x^i$$
or basically
$$ 1 + 2x + 3x^2 + 4x^3 +... ...
3
votes
2answers
214 views
Monotonicity of $\ln n\over \sqrt{n}$
Is it possible to prove the monotonicity of $(a_n)_n =\frac{\ln n}{\sqrt n}$ without using derivatives?
0
votes
1answer
49 views
Applying Binomial theroem
Using the equation; $F_{r}=12\frac{U_{0}}{R_{0}}\left[\left(\frac{R_{0}}{R_{0}+x}\right)^{13}-\left(\frac{R_{0}}{R_{0}+x}\right)^{7}\right]$
I must apply the binomial theroem to; ...
3
votes
3answers
175 views
the sum of a series
I am stuck on the computation of the following sum:
$$\sum_{k=0}^{\infty} {\Big( {\frac{q}{k+1}} \Big)}^k ,$$
where $k$ is a natural number, and $0<q<1$.
2
votes
1answer
105 views
Solving a series $n(1 + n + n^2 + n^3 + n^4 +…n^{n-1})$
I'm trying to sum the following series?
$n(1 + n + n^2 + n^3 + n^4 +.......n^{n-1})$
Do you have any ideas?
7
votes
1answer
227 views
Solving a formal power series equation
I want to find a function $f(x,y)$ which can satisfy the following equation,
$$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = \exp \left[ \sum _{n=1} ^\infty ...
3
votes
4answers
219 views
The exact value of $\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$
What is the value of :
$$\sum_{n=1}^{\infty}\frac{n^2+n+1}{3^n}$$
5
votes
1answer
159 views
When does the summation of a quotient equal the quotient of summations?
That is, under what conditions would
$$
\sum_{i = 1}^n \frac{a_i}{b_i}= \frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i}
$$
be true? What about for infinite summations, i.e. when $n \rightarrow ...
0
votes
0answers
39 views
Finding an expression for a given series
Is there a problem on basic complex analysis, i need to find an expression for the result of the finite sum
$ \sum \limits_{n=0}^{M-1} e^a$, where $a=(i \frac{2\pi}{M{}}nk)$, this result must depend ...
2
votes
4answers
222 views
Calculating the chess problem, sum $\sum_{k=0}^{63}2^{k}$ [duplicate]
Possible Duplicate:
Summation equation for $2^{x-1}$
I'm solving the classic problem of the inventor of chess, who according to legend sold the invention for one grain for the first square ...
0
votes
0answers
115 views
How to prove that $(1^3+2^3+\cdots+n^3)=(1+2+\cdots+n)^2$ [duplicate]
Possible Duplicate:
Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$
How can one prove that ...
11
votes
3answers
286 views
Collatz-ish Olympiad Problem
The following is an Olympiad Competition question, so I expect it to have a pretty solution:
For a positive integer $d$, define the sequence:
\begin{align}
a_0 &= 1\\
a_n &=
...
1
vote
4answers
426 views
Sum of n different positive integers is less than 100. What is the greatest possible value for n?
I am stumped on the following question:
The sum of n different positive integers is less than 100. What is the greatest possible value for n?
a) 10, b) 11, c) 12, d) 13, e) 14
...
2
votes
2answers
43 views
Should we consider multiplicity while solving this problem?
I am trying to solve the problem :
A single fence is to be constructed from posts 6 inches wide and separated by lengths of chain 5 feet . If a certain fence begins and ends with a post.Which ...
2
votes
2answers
249 views
Does this qualify as a proof? (Spivak's 'Calculus')
I'm working through Spivak's 'Calculus' at the moment, and a question about series confused me a bit. I think I have the solution, but I'm not sure if my "proof" holds.
The question is:
Prove ...
