You mean the way i proved $$\lim_{n\rightarrow +\infty}\frac{n}{1 + n(a_n)} = \infty$$H
How about this:
$$\lim_{n\rightarrow +\infty}\frac{n}{1 + n(a_n)} = \lim_{n\rightarrow +\infty} n * \lim_{n\rightarrow +\infty} \frac{1}{1 + n(a_n)}=\infty*1=\infty$$. I have $$\lim_{n\rightarrow...
Homework Statement
Let $$(a_n)$$ be a sequence such that $$\lim_{n\rightarrow +\infty}n(a_n)=0$$.
1) What is
$$\lim_{n\rightarrow +\infty}(1 + {\frac{1}{n}} + (a_n))^n$$
2) For which value of p and l, after some n is $$(b_n)=\frac{n^{p \ cos(n\pi)}}{(1 + l + (a_n))^n}$$ properly defined. p...
I don't think that motivation will make you work around the clock, work ethic will. I once heard someone along the lines: Motivation gets the ball rolling, work ethic/discipline keeps the ball rolling. That being said i think best way to develop work ethic is to develop work habits. Since you...
Homework Statement
Let f: R -> R and defined with f(x)={ x, if x \in Q or x^2 if \in R\Q}
a) Prove that function is discontinuous at x=2;
b) Find all points for R in which function is continuous;
The Attempt at a Solution
As far as i know there are infinitely many irrationals but more...
Since you gave counter example you assumed that S is an order set with least-upper-bound property. (Right?) If so then this means that if E is subest of S, and E is non empty and E is bounded above , then sup E exists in S.
So let S = (0,1) ##\cup## (2,3) and Let E= (2,3). E is subset of S...
Theorem: Suppose S is an ordered set with the least-upper-bound property, B⊂S, B is not empty, and B is bounded below. Let L be the set of all lower bounds of B. Then α=supL exists in S, and α=infB.
Rudin proves that α=supL, α is an element of L and that α=infB.
For α to be sup i.e. lub it...
Homework Statement
If p is a prime and k is an integer for which 0<k<p, then p divides \displaystyle \ \binom{p}{k}.
Whne p divides \displaystyle \ \binom{p}{k} it means that \displaystyle \ \binom{p}{k}=p*b.
wheren b is some number.
The Attempt at a Solution
So p is equal to some number k...
The question was: How many real number solutions are there for 2^x=-x^2-2x. I tired for an hour to isolate x but i couldn't do it. Then i used wolfram alpha and it gave me two solutions and graph. I realized that question was, how many not what are the solutions, and i could do that by graphing...
Homework Statement
Numbers a,b,c are consecutive members of increasing arithmetic progression, and numbers a,b,c+1 are consecutive members of geometric progression. If a+b+c=18 then a^2 +b^2 + c^2=?
The Attempt at a Solution
a + b + c= 18
a + a +d +a + 2d = 18
3a + 3d = 18
3(a+d)= 18...
As i have understood it there exist separate arithmetic and separate geometric progression. This is a first time i hear of arithemtico-geometric series.
Homework Statement
If a,b,c, are at the same time fifth, seventh and thirty seventh member of arithmetic and geometric progression then a^{b-c}b^{c-a}c^{a-b} is:
The Attempt at a Solution
I tried solving system of equations but i have four unknown. I was able to reduce it to on unknown...
Homework Statement
Let X be a set containing all four digit numbers made up of {1,2,3}, where every number contains every digit at lease once. Number of all subsets is:
The Attempt at a Solution
So firs i have to find number of elements in the set:
3!*3 + 3*12 = 54
Now what they...
So you mean to use Spivak's textbooks instead of Abbott or to just add Spivak to the mix?
I know that Rudin doesn't have great rep for user friendliness. I thought using Abbott first as an intro and then to study Rudin.
In October of this year i will start with math major and i decided to prepare myself in spear time. In my faculty there is no such thing as Calculus but rather you go strait to the analysis and you pick up calculus along. (There is singe variable calculus in high school). In first two years...
Is there planned research that tries to create new organ for transplantation? So when someone needs a new organ e.g. kidney, instead of finding a donor, scientists could create an organ form stem cells perhaps. Is this even psychically possible?
Reason i ask this question is that i watched...
Well i know that center is at x=3.
I know that the slope between A and B is -√3 (3-3-√3)/(1)
If i want to find a line that goes throug the points it should be like this:
For A:
y-0=-√3(x-3)
y=-√3x + 3√3
but for B i get different line
y+1=-√3(x-3 -√3)
y=-√3x + 3√3 +2
Two different lines...
Homework Statement
Find the equation of a cricle if it touches x-axis in A(3,0) and it contains B(3 + √3, -1)
The Attempt at a Solution
Is there a piece of data missing here? Because i can't see how can i find center of circle.
Maybe i will try to slove it,
This problem should be difficult according to my textbook, here is a pic from my textbook:
http://img444.imageshack.us/img444/8316/img00121201304241927.jpg [Broken]
That is exactly what i need to prove, i just didn't know how to write it like that using Latex. Here is full text of problem:
Prove that \displaystyle \ \binom{n}{m}=\binom {n} {m-2}+2\binom {n-1} {m-1}+\binom {n-2} {m-2}\ n\geq2, m \geq2
When i plug in n=m=2 i get \frac{0}{0} =...
After that i get (n-2)!*(n(n-1)(m-1) + 2(n-m+2)(n-m+1)(n-1) + (m-1)(n-m+2)(n-m+1))/(m-1)!(n-m+2)!
i tried and i can't find any other common factor in numerator and i have common factors between two products such as (n-1) , (m-1), (n-m+2)(n-m+1). Only other way is to multiply everything and then...
If i want get whole expression under one fraction i should multply first fraction with (m-1)/(m-1) to get n!(m-1)/(m-1)!(n-m+2)! , second fraction with (n-m+2)(n-m+1)/(n-m+2)(n-m+1) to get
(n-1)!(n-m+2)(n-m+1)/(m-1)!(n-m+2)! , and third with (m-1)(n-m+2)(n-m+1)/(m-1)(n-m+2)(n-m+1)
to get...
Homework Statement
Prove the following:
\frac{n!}{m!(n-m)!} = \frac{n!}{(m-2)!(n-m + 2)!} + 2* \frac{(n-1)!}{(m-1)!(n-m)!} + \frac{(n-2)!}{(m-2)!(n-m)!}
The Attempt at a Solution
I tried writing following
\frac{n!}{m!(n-m)!} = \frac{n!}{m!(n-m)!}(\frac{m(m-1)}{(n-m + 2)(n-m +...
Homework Statement
1.) 4 of the books have red covers, 3 have green covers, and another 2 have gray covers. In how many ways can the books be arranged on a shelf if books of the same color must be arranged together?
The Attempt at a Solution
1) I think that the answer here is 3...
Homework Statement
Find parameter a so that line y=ax + 11 touches ellipse 3x^2 + 2y^2 = 11
The Attempt at a Solution|
I can rewrite ellipse equation like \frac{x^2}{\frac{11}{3}} + \frac{y^2}{\frac{11}{2}} = 1
And i know that line y=kx + n touches ellipse when a^2k^2 + b^2 = n^2...
Homework Statement
Show that \frac{a+b}{2}\geq\sqrt{ab} for 0 < a \leq b
The Attempt at a Solution
Since b \geq a then b + a \geq 2a and 2b\geq a + b
That is \frac{a+b}{2}\geq a and \frac{a+b}{2}\leq b.
Since \frac{a+b}{2}\leq b can i multiply \frac{a+b}{2}\geq a with b?
If...
Hello friends.
It seems to me that i have reach a wall with my math education. In 3.5 months i will be doing tests for mathematics faculty(college), something like SAT but a bit more difficult i think.
Anyway i have been doing more difficult problems from problem set and i simply can't figure...
Homework Statement
I should sketch function (x^2 + x -12)/(x-4).
The Attempt at a Solution
I have problem with first derivative i find it to be (x^2 - 8x +8)/(x-4)^2 with roots at 4 - 2√2, 4 + 2√2, and 4(we lose four because f is not defined at 4). Where at 4 - 2√2 f is at max and...
I have been using physics forum for good year and a half now for homework problems and for asking for advice, so i wanted to subscribe but it is only possible to do it over paypal, but in Serbia where i live we don't have paypal so i can't do it. Is there any other way to subscribe and if not it...
Exactly that kind of the system is not random, for me truly random system is system that you can't possibly predict, even if you have all the possible data and know all the laws of physics.
Hi everybody,
i didn't wanted to create two separate threads so merged them into one.
i got confused watching Brian Green explaining QM on one of his shows.
He compared distribution in double slit experiments with throwing a ball on a roulette.
He said that casino doesn't have to know...
Homework Statement
Let f be defined for all real x and suppose that,
\left|f(x) - f(y)\right|\leq(x-y)^2
for all real x and y. Prove that f is constant.
The Attempt at a Solution
First of all, is following allowed. Since f is constant then \left|f(x) - f(y)\right|=0, and form here to build...
Homework Statement
Of all cylinders inscribed in sphere of radius R largest area of side(M) has cylinder which hight is R\sqrt{2}. Prove.
The Attempt at a Solution
I understand how to prove this i only have problem with derivative:
M=2*r*Pi*H and r=\frac{\sqrt{(2R)^2 - H^2}}{2}...