Andrew Baxton

1 + 2 + 3 + .... + i - 2 + i - 1 = [( i - 1)i]/2

Prove with gory detail please.

Let S be a set. Prove that the law of composition defined by ab = a for all a and b in S is associative. For which sets does this law have an identity?

Show by example that the product of elements with finite order in a group need not have finite order. What if the group is abelian?

I think it's something like g^k = g^s

Then (g^k)(g^-k) = (g^s)(g^-k) = g^s-k = e

Therefore, there is a smallest integer m = s-k s.t g^m = e.??

a) Prove that the elementary matrices of the first and third types (1,2,3) generate GL_n(R)

b) Prove that the elementary matrices of the first type generate SL_n(R). Do the 2x2 case first.

Let x and y be elements of a group G. Assume that each of the elements x,y and xy has order 2. Prove that the set H = {1,x,y,xy} is a subgroup of G, and that it has order 4.

I know it has order 4 because it has 4 elements, but the subgroup is throwing me off.

a) Let G be a cyclic group of order 6. How many of its elements generate G? Answer the same question for cyclic groups of orders 5 and 8.

b) Describe the number of elements that generate a cyclic group of arbitrary order n.

Let a and b be elements of a group G. Prove that ab and ba have the same order.

Let a and b be elements of a group G. Assume that a has order 7 and that a^3b = ba^3. Prove that ab = ba.

x1 + x2 + 2x3 - x4 = 3

Find the sum n=1 -> infinity of (-1)^(n-1)/n^2 correct to 2 decimal places.

The next thing my prof. put was "i.e. need Rn < 0.005"

I don't understand how he got the 0.005...please explain clearly, have an exam tonight.

Integral of x - 4 / x^2 +2x +5....steps please.

Integral of sqrt((6t)^2 + (6t^2)^2) | 0->1

=sqrt(36t^2 + 36t^4)

=6*sqrt(t^2 + t^4)

6(outside integral)=t*sqrt(1 + t^2)

let t = tanO {where O is theta)

then sqrt(1 + tan^2O) = secOdO

6=tanOsecOdO

let u = tanO du = secOdO

=6u

=6/2u^2

=3u^2

=3tan^20

since tanO = t, then...

=3t^2 | 0->1

=3 - 0 = 3