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1. ### Abstract Algebra: Show that 2Z + 5Z = Z

The TA said that we're supposed to approach the problem as to show 2Z + 5Z = {a+b | a in 2Z and b in 5Z} = Z. I forgot to paste that into #2. Other than that, you have 2Z and 5Z right. I'm sorry. I was in a rush to get to class and I didn't take my time typing out the post. I understand where...

You need to clean up your notation a little bit, but the ROC for this series is easily found using any number of tests. Try the ratio test or the limsup test.
3. ### Abstract Algebra: Show that 2Z + 5Z = Z

Homework Statement Show that 2\mathbb{Z} + 5\mathbb{Z} = \mathbb{Z} Homework Equations where 2Z + 5Z = {a+b | a in 2Z and b in 5Z} = Z The Attempt at a Solution For any n in Z, we can write n= (5-4)n = 5n +(-4)n = 5n + 2(-2n) And since 5n is in 5Z and 2(-2n) is in 2Z, we can form Z from...
4. ### Show a function is differentiable everywhere, and show its derivative is continuous

Homework Statement show f(x)=\left\{e^{\frac{-1}{x}} \\\ x>0 f(x)=\left\{0 \\\ x\leq 0 is differentiable everywhere, and show its derivative is continuous Homework Equations Product Rule and Chain Rule for derivatives. Definition of a derivative f^{'}(a)=\frac{f(x)-f(a)}{x-a}...
5. ### Find the limsup and liminf

Homework Statement find the limsup and liminf of a_{n}=(-1)^{n}\left( 1+\frac{1}{n} \right) Homework Equations Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written lim sup is the limit of a_{n}=sup_{(k>=n)}a_{k} The...
6. ### If a given permutation in S_n has a given cycle type, describe sgn(sig).

Yup. But since you can write any n-cycle as a product of 2-cycles, how do I account for the cycles of odd length.
7. ### Show the Symmetric group is generated by the set of transpositions (12) (n-1 n).

Oh, my bad. I wrote the set down wrong. WTS: Any 2-cycle is generated by the set {(12)(23)....(n-1 n)} So taking any 2-cycle (i j) where j > i (I'm going to work on the difference between j and i. I guess what follows is a sketch. I'll clean it up later.) If j-i=1, then (i j) = (i i+1)...
8. ### For (x,y) in R2, describe the set of orbits geometrically.

Homework Statement 4a) If x, y are in R, prove that (R, +) acts on R2 by (x,y)*r = (x+r, y) for all (x,y) in R2 and for all r in R. b) If (x,y) are in R2, find the orbit of (x,y). Describe geometrically. Homework Equations none that I can think of The Attempt at a Solution The...
9. ### Show the group action on (x, y) and, desribe the orbits.

Homework Statement 1 a) Prove that the group (nZ, +) acts on Z by a*g = a + g for all g in nZ and for all a in Z. b)What are the orbits? c)How many orbits are there? Do the set of orbits remind you of anything in number theory? Homework Equations not sure The Attempt at a Solution...
10. ### If a given permutation in S_n has a given cycle type, describe sgn(sig).

Homework Statement 5.4: If sigma in S_n has cycle type n_1,...,n_r, what is sgn(sig)? (sgn is the sign homomorphism) Homework Equations sgn(sigma) = 1 if sigma is even. sgn(sigma) = -1 is sigma is odd cycle type is the length of the cycle type. If n_2 = 2, sigma has two 2-cycles. The...
11. ### Show the Symmetric group is generated by the set of transpositions (12) (n-1 n).

Homework Statement 5.1: Prove that S_n is generated by the set {(1 2), (3 4),...,(n-1 n)} Homework Equations None that I know of The Attempt at a Solution Any element in S_n can be written as a product of disjoint n-cycles. So now I need to show any n-cycle can be written as a product of...