Inequality Definition and 1000 Threads

Homework Statement Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that \frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} < \frac{1}{2} Homework Equations cosx = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \cdot \cdot \cdot = \sum_{n=0}^\infty...

Homework Statement Prove that if n is a natural number and is greater/equal to 4, then 2n<n!, and show that 2n is less/equal to 2((n-1)!) follows. The Attempt at a Solution I'm thinking I just need to use induction on n for the first part, where I get the inequality (n+1)! = n!*(n+1) >...

Homework Statement d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2} where a>0, b>0, c>0 and 4ac-b^2<0 Show whether d(x,y) exhibits Triangle inequality? Homework Equations (M4) d(x,y) \leq d(x,z)+d(z,y) (for all x,y and z in X) The Attempt at a Solution I...

Homework Statement Show that 1/2(1-cos1)\leq\int\intsinx/(1+(xy)4)dxdy\leq1 on the area 0\leqx\leq1, 0\leqy\leq1. Homework Equations Mean Value Inequality: m*A(D)\leq\int\intf(x,y)dA\leqM*A(D), where m is the minimum and M is the maximum on the interval. The Attempt at a Solution...

Homework Statement Use your knowledge of exponents to solve \frac{1}{2^x} > \frac{1}{x^2} Homework Equations The Attempt at a Solution x^2 > 2^x Then I am stuck. I know they intersect at x = 2.

Homework Statement 1/x <= 4 Homework Equations The Attempt at a Solution I initially converted 1/x back to x^-1 which gave me the answer x <= 1/4 which makes sense, but I should also get x < 0 which I'm not sure about how to get via solving? Also is converting 1/x to x^-1 the...

Suppose f, g:[a,b]->R are bounded & g(x)<=f(x) for all x in [a,b] for P a partition of [a,b], show that L(g,P)<=L(f,P) I don't know whether I should show by cases since I don't know the monotonicity of the both functions f and g. It seems like that the graphs of both functions have to behave...

Hi everyone Today during problem session we had this seemingly simple exercise, but I just can't crack it: We should give an example of an x \in \ell^2 with strict inequality in the Bessel inequality (that is an x for which \sum_{k=1}^\infty |<x,x_k>|^2 < ||x||^2, where (x_k) is an orthonormal...

Homework Statement prove that for all x>0 Homework Equations -1 \leq sin t \leq 1 The Attempt at a Solution the area under the graph is increasing as x increases also, i tried to write it the sigma way: then take the limit as n-->infinity i got stuck trying to figure out how to...

Homework Statement For what values of the variable x does the following inequality hold: \frac{4x^2}{(1-\sqrt{1+2x})^2}<2x+9 Homework Equations The Attempt at a Solution Maybe some hints for me to begin.

I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?

Homework Statement If x is real and y=\frac{x^2+4x-17}{2(x-3)}, show that |y-5| \geq 2Homework Equations The Attempt at a Solution Sorry... Absolutely no idea. I tried to substitute y into the left side to prove that -2 \leq y - 5 \leq 2 but I can't. Anything I should know to do this?

Homework Statement P(AUB) <= P(A) + P(B) Homework Equations The Attempt at a Solution I can't understand the intuition behind this property. It's not a homework assignment, it was just something that came up in class. Thanks, M

Homework Statement If a, b and c are distinct positive numbers, show that 2 (a^3 + b^3 + c^3) > a^2b + a^2c + b^2c + b^2a + c^2a + c^2b Homework Equations The Attempt at a Solution I have tried to expand from (a+b+c)^3 > 0, also tried (a+b)^3 + (b+c)^3 + (c+a)^3 > 0, and then...

Suppose I have three vectors a,b and c in R^d , And, I have that a.b < c.b(assume Euclidean inner product). What are the ways to visualize relation between a,b and c geometrically? I realize this is slightly open-ended, but am looking for insight here. Thanks in advance. PS: I have a thought...

I expanded (x+y),(x+y) and got x^2+y^2 > 2xy then replaced 2xy with 2|x,y| but now I'm stuck. I need to get it to ||x+y|| <= ||x|| + ||y||. Am I close?

Homework Statement A rectangle solid is to be constructed with a special kind of wire along all the edges. The length of the base is to be twice the width of the base. The height of the rectangular solid is such that the total amount of wire used (for the whole figure) is 40cm. Find the...

Homework Statement Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality (1+R_{G})^{n} \leq V is true. Where R_{G} = (r_{1}r_{2}...r_{n})^{1/n} and V= \Pi_{k=1}^{n} (1+r_{k}) Homework Equations The Attempt at a Solution I've...

Homework Statement Show that \forall a,b \in R: \left|ab\right|\leq\frac{1}{2}(a^{2}+b^{2}) Homework Equations Triangle Inequality seems to be useless. The Attempt at a Solution (a+b)^{2}=a^{2}+b^{2}+2ab 2ab=(a+b)^{2}-(a^{2}+b^{2})...

Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...

Hi, It is well known by Varignon's Theorem and the Triangle Inequality that in a convex quadrilateral ABCD with midpoints a of side AB, b of side BC, ... d of side DA and perimeter p, \[ AC + BD < p\] where AC and BD are the diagonals of the quadrilateral. However, how do I obtain...

Hi, It is well known by Varignon's Theorem and the Triangle Inequality that in a convex quadrilateral ABCD with midpoints a of side AB, b of side BC, ... d of side DA and perimeter p, AC + BD < p where $AC$ and $BD$ are the diagonals of the quadrilateral. However, how do I obtain...

Pretty much knows the triangle inequality. \left| a + b \right| \le \left| a \right| + \left| b \right| I was reading a source which asserted the following extension of the triangle inequality: \left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right) This is...

hi every body show if Xn→x then lXnl→lxl hint use trangle inequality 2/ show if lXnl→0 then Xn →0 show by example that lXnl fore all n in N MAY CONVERGE and Xn may not converge

Homework Statement prove that llal-lbll\leqla-bl Homework Equations Triangle inequality lx+yl\leqlxl+lyl The Attempt at a Solution Let a=(a-b)+b By using the triangle inequality we get lal-lbl\leqla-bl Then from here I am not sure what I can do. I would like to say on the left...

Homework Statement Prove a)5 < 51/2 + 51/3 + 51/4 b) n > n1/2 + n1/3 + n1/4 for all ints n>8 Homework Equations The Attempt at a Solution i tried the AM GM inequality and found 51/2 + 51/3 + 51/4 > 3(513/36) what further can i do? can anyone please help me out??

Homework Statement The question says to find a proof for Cauchy's Inequality and then the Triangle Inequality. This is an elementary linear algebra class I'm doing, so I can't use inner products or anything. Homework Equations The Attempt at a Solution I got the proofs using algebra, but I'm...

Homework Statement Show \sum_{i=1}^n \frac{1}{i^2} \leq 2 - \frac{1}{n} with induction on n. I'm pretty rusty on induction (not that I was very good at it to being with), so I mostly wanted to know if I'm on the right track, and if this is a way towards a valid proof. Homework Equations The...

Homework Statement Show that if n belongs to N, and: An: = (1 + 1/n)^n then An < An+1 for all natural n. (Hint, look at the ratios An+1/An, and use Bernoulli's inequality) The Attempt at a Solution I think i have a vague idea of what to do here, like I am sure induction is involved in this...

Homework Statement Prove that 1 + 1/2 + 1/4 + 1/7 + 1/11 + ...... <= 2*pi Homework Equations none The Attempt at a Solution all i could figure out was the nth term of the sequence T(n) = \frac{2}{2 + n(n-1)} any help appreciated.:biggrin:

Homework Statement solve the following equations or inequalties for x in the interval [0,2pi) 2cos^2(x) + 1 = 3cos(2x) Homework Equations The Attempt at a Solution My attempt at the problem: 2cosx(cos2x) + 1 = 3cos(2x) 2cosx(cos^2x - sin^2x) + 1 = 3cos(2x) 2cos^3x -...

How can we graph this inequality - |y|+1/2>=e-|x| ? I drew the function(actually a combination of functions) for equality. It would be symmetric in all quadrants and intersect the axes at +ln2 and -ln2(x-axis) and 1/2 and -1/2(y-axis).However since the various graphs are mixed up it is hard...

Inequality Solution [URGENT, +Part solution included] Homework Statement l 3/(x-1) - 5 l < 4 Homework Equations The Attempt at a Solution so here's where I am abit confused. since the inequality sign is not > or >= but instead in this case it is <. Therefore, x has to be...

Homework Statement Given:0 <= a <= b a <= Sqrt(ab) <= (a+b)/2 <= b Homework Equations The Attempt at a Solution The only problem I am having prooving this inequality is Sqrt(ab) <= (a+b)/2. I have an idea but I am not sure if it validates. can i do this.. ? (a+b)/2 - sqrt(ab) >= 0 if it is...

Homework Statement Let x and y be real numbers. Prove that if x =< y + k for every positive real number k, then x =< y The Attempt at a Solution x =< y + k -y + x =< k since k is positive, the lowest value it can take doesn't include 0: -y + x < 0 x < y So I get x < y from x =< y...

Homework Statement l [3/(x-1)] - 5l < 4Homework Equations The Attempt at a Solution My 1st step was to make the inequality like this. -4 < 3/(x-1) - 5 < 4 and then i multiplied (x-1) to both left and right side and as well as to the 5. but in the end, my result turns out to be...

Homework Statement Given 0 <= a <= b show that, a <= sqrt(ab) <= (a+ b / 2) <= b Homework Equations a * b <= a^2 / a*b <= a* a The Attempt at a Solution I think i know where I am going but i wanted to make sure if its correct so far. So we know that...

Homework Statement 1] l x + y l < or equal to l x l + l y l Homework Equations x^2 + 2xy + y^2 The Attempt at a Solution Left side. i Squared left side to begin with, and i got x^2 + 2xy + y^2 and also did the same for the right side, but it would have absolute sign...

Homework Statement From Spivak's Calculus, Chapter 2 Problem 22 Part A: Here, A_{n} and G_{n} stand for the arithmetic and geometric means respectively and a_{i}\geq 0 for i=1,\cdots,n. Suppose that a_{1} < A_{n}. Then some a_{i} satisfies a_{i} > A_{n}; for convenience, say a_{2} >...

I know that we can square an inequality if both sides are positive. But can we cube an inequality provided both the sides are positive? If no then why?

Homework Statement *Sorry I could not get the math symbols to work properly so I did it by hand. I hope this isn't too much trouble. Prove: | sqrt( x ) - sqrt( y ) | <= | sqrt ( x - y ) | for x, y >= 0 Hint: Treat the cases x >= y and x <= y separately. I am new to proofs and we can't use...

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|. How do we know that? is |z+w|=|z-w|?? Note that z and w are complex numbers.

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|. How do we know that? is |z+w|=|z-w|?? Note that z and w are complex numbers.

given z, w\inC, and |z|=([conjugate of z]z)1/2 , prove ||z|-|w|| \leq |z-w| \leq |z|+|w| I squared all three terms and ended up with : -2|z||w| \leq |-2zw| \leq 2|z||w| I know this leaves the right 2 equal to each other but i figured if i show that since there exists a z\geqw\geq0, then...

My problem states: Given z, w\inC, prove: ||z|-|w||\leq|z-w|\leq|z|+|w|. Now, I am confused because, isn't it true that ||z|-|w||=|z-w| ? I am using Rudin's book which gives |z|=([z's conjugate]z)1/2

Homework Statement -3<(1/X)≤ 1 Solve. Homework Equations The Attempt at a Solution Here's my attempt at it: (1/X)≤1 and (1/X)> -3 X≥ 1 and X> (-1/3) Am I doing something wrong here? Is this the complete solution? Looking at my answer, is there more that I...

Hi everyone, Homework Statement P(X)=Xp-1*(X-1)p*...*(X-n)p j is an integer between 1 and n; x a real beatween 0 and 1. Prove that abs(P(jx))<=(n!)p Homework Equations The Attempt at a Solution I tried to find an inequality for each abs(jx-q) but the problem is that I...

Homework Statement 2 - ((x-3)/(x-2)) ≥ ((x-5)/(x-1)) Homework Equations The Attempt at a Solution I just want to make sure that certain operations that are allowed with fractions in equations are still valid (or not) in inequalities. 2 - ((x-3)/(x-2)) ≥ ((x-5)/(x-1)) Can I...

Homework Statement l lxl - lyl l =< lx-yl Homework Equations n/a The Attempt at a Solution how do i proof this? give me a start please, should i use definition absolute values and consider all of the cases? or use triangle inequality(but i can't figure out how)

Say I have two lists, List1 and List2 containing elements such as words. Some words are common two both List1 and List2. I want to create a distance metric that tells me how far apart the two lists are based on a similarity "score". The similarity score and distance metric are as follows...