Recent content by Mathematicsresear

The definition I have in mind is as follows: For all M>0 there exists an N in the natural numbers such that for all n >= N, x_n <= -M I am aware that I can say that a sequence of the form (-n)/c where c is a positive integer tends to negative infinity as n grows larger, by dividing the top and...

Yes, I am aware of these relations, however, I'm trying to prove that the sequence diverges from the definition.

I chose N to be the maximum of 3 and 21M/5

What if I defined M to be greater than or equal to 0 in the proof? I can do that since that's an alternative definition for the divergence to negative infinity, right?

Homework Statement prove (5-n^2)/(3n+1) diverges to negative infinity as n approaches infinity Homework Equations For all M>0 there exists an N in the natural numbers such that for all n >= N, x_n <= -M The Attempt at a Solution Let M be an element of the field of the real numbers. Let N in...

<Moderator's note: Moved from a technical forum and thus no template.> 1. Homework Statement Is this proof correct? Let K>0, and choose N such that N >= K2, then for all n in the naturals, and n>=N, sqrt(n)+7>=sqrt(N)>=K Is this proof correct? Please tell me

Homework Statement Suppose sequence x_n tends to 0 as n approaches infinity, prove that sqrt(x_n) also tends to 0 x_n is a sequence of non negative real numbers Homework EquationsThe Attempt at a Solution Proof. Let e>0. There exists an N in the naturals such that for n>N Ix_nI < e So if I...

Alright, I understand. The gradient if perpendicular to a surface, but it also is pointing in the direction of greatest increase, I'm not sure what the link between those to are. So is it both, pointing in the direction of greatest increase, and perpendicular to the surface?

I mean, would it also be perpendicular to the function g(x,y,z)?

<Moderator's note: Moved from a homework forum.> 1. Homework Statement Is the gradient perpendicular to all surfaces or just level surfaces? For instance, if I I have a function f(x,y)=z where z is the dependent variable then that is a surface, wouldn't that be a level surface to a function of...

Homework Statement I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question? Standard basis vectors: e_1, e_2 or i,j

no

Homework Statement Prove sequence ((-1)^m)m diverges. Homework Equations for all epsilon greater than zero, there exists a natural number M such that for all natural numbers m greater than or equal to M, Ix_m-xI is less than or equal to epsilon.[/B]The Attempt at a Solution Assume it...

Homework Statement Why is it that the probability of getting a queen in my second draw given that the first card was a spade, independent events? What if the first card drawn was the queen of spades? Homework Equations P(A and B)=P(A)P(B)

What about my other comments? because that's exactly what I'm confused about, why is atleast turned into an existential quantifier in the beginning of the sentence but not in the other part of the sentence?