# proof

1. ### functions, relations, injective,...

Hello there, I'm new to the kind of math you have to prove, so be patient ^^ Are there any improvements I could make here? In the following, I will talk about $Z$ as a ring. More specifically I consider $(Z,+,\times)$ a commutative and unitary ring, where $Z = \{...,-2,-1,0,1,2,...\}$. I...
2. ### Proof of sample variance formula

This formula should be checked for i+1: s_{i}^2= \frac{1}{i-1}\sum\limits_{k=1}^{i} (x_k-\bar{x}_i)^2 The result should be this: s_{i+1}^2=\left(1-\frac{1}{i}\right)s_{i}^2+(i+1)(\bar{x}_{i+1}-\bar{x}_{i})^2 Assumed as known, this is...
3. ### proof

abc and bca are 3 digit numbers.show that abc-cba is divided by 11
4. ### Easy proof of 22/7>pi

Let x=22/7-\pi\; and suppose 22/7<\pi or x<0. Since x is negative then sgn(x)=-1+2H(x)<0. sgn(x)=-1+2\lim_{s\rightarrow \infty} \frac{1}{1+e^{-sx}}=-1+2\lim_{t_s \rightarrow \infty} \frac{1}{1+e^{t_s x}}. Since the limit of 1+e^{-sx} converges then : sgn(x)=-1+2=1 which is a contradiction...
5. ### Proof for a challenging inequality

I have something I believe to be true, but I'm uncertain, so I'm looking for a proof. For positive real numbers a,b,c,d Prove that if a>=b and c<=d, then a/c <= b/d
6. ### Help with a proof

Hey guys. I found out that a Î£2n n=1 EQUALS [a^2 - a] However, although I can see that it works, I can't figure out why this is true. Can anyone figure this out?
7. ### Proof for square root series

https://ibb.co/LvJkVGZ In this image I have a sum series and I need to proove that all the terms from the LHS are equal to the two terms from the RHS. I wrote a solution but I'm not 100% certain this is correct. https://ibb.co/0fmVq3y
8. ### Induction proof

Prove inequality with induction :\prod_{k=1}^{N} k^{k} <2e^{- N^{2}}N^{6N^{2} +6N+1}.
9. ### Help with a geometric proof

Let ABC be a triangle with AC>BC. Let M be the midpoint of the segment AB, D the intersection point of the line AB with the angle bisector of BCA, L the foot of the height through C. Let further E be the touching point of AB and the incircle of the triangle ABC. Let the tangent to the incircle...
10. ### Find all a such that n^aâˆ’n is divisible by aâ‹…(aâˆ’1) for any integer n.

 $$a \cdot (a-1) \ \mid \ n^{a}-n \ \ \ \forall \ n \in \mathbb{Z}$$ Find all $a$. 
11. ### Proof of e^x, lnx and e as limit without circular dependency?

People prove the derivative of e^x, \ln(x) using either the formula of e = \lim_{x \to 0} (1+x) ^ \frac{1}{x} or if they know one of the derivatives e^x, \ln(x) they use it to prove the other. My textbook first gave me without proof that \frac{d}{dx}e^x = e^x Then it found the limit of \ln(x)...
12. ### Proof of goldbach's conjecture

PROOF OF GOLDBACHâ€™S CONJECTURE Goldbachâ€™s conjecture states that every even integer > 2 is the sum of 2 primes. An example is 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53. An alternative statement of the conjecture is as follows. Every integer > 3 is the arithmetic mean of...
13. ### proof of property

Let a 3 Ã— 3 matrix A be such that for any vector of a column v âˆˆ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
14. ### Help with proof?

The givens are, Quad ABCD is inscribed inside the circle Diagonals AC and BD meet at point E. Line AD is congruent to Line CD How do you prove that BE * AD = EC * AB?
15. ### Inequality proof

Prove : 1+1/2 +1/3 +....+ 1/2^{k} \geq 1+k/2. For k=0,1,2,3... .
16. ### Asymptotic Normality proof

I'd like some assistance on the proof of the following Lemma: If $X_n$ is $AN(\mu,\sigma_n^2)$, then also $X_n$ is $AN(\overline\mu,\overline\sigma_n^2)$ if and only if \frac{\overline\sigma_n}{\sigma_n}\rightarrow1, \frac{\overline\mu_n-\mu_n}{\sigma_n}\rightarrow0. The hint says to use...
17. ### Proof for natural logarithm limit without differentiation

I have another limit where I'm struggling with the solving.Can somone help me? https://ibb.co/5Kq58jR I used a method like this but I'm not 100% if it's correct! https://ibb.co/JQrpdL1 What should I do from that step? I know something like arctan(x)/x when x tends to 0 is 1. How is that...
18. ### Proof for this power limit without differentiation

https://ibb.co/WVC8BgJ I am stuck at the last step and I don't know how to go on.Can someone help me?What do I do from the last step?
19. ### proof

How can I proof (Kn, t) = t(t 1)(t 2)Â·Â·Â·(t (n 1)). number of way of coloring a complete graph (Kn) with n vertex, with palette of t colors.
20. ### combinatorial proof

How can I give combinatorial proof to this ?