Series convergence

Prove that the series given by the recurrence relation a_n+1 = SQRT(3*a_n), where a_1 = 4, converges, and find the limit of convergence.

Multiplicative groups

(See attached file for full problem description) --- a) Recall the following definitions of the multiplicative groups GLn(k) and SLn(k) over a field k: GLn(k)={invertible n x n matrices over k} SLn(k)={A in GLn(k) such that the determinant of A=1} Prove that SLn(k) is a normal subgroup of GLn(k) and that the quotient group GLn(k)/SLn(k) is isomorphic to the multiplicative group k*={a in k such that a is not equal to zero}. b) Determine the number of elements in the finite group GL3(Zp)

Find the orbit and stabilizer of the given matrix under multiplication by matrices of the specified type.

Find the orbit and stabilizer of the 2 X 2 matrix M under the action of multiplication of M by the matrices in GL_2(R), where the top row of M is (1 0) and the bottom row is (0 2). [That is, m_11 = 1, m_12 = 0, m_21 = 0, and m_22 = 2.] See attached file for full problem description.

Eigenvalues, eigenfunctions, modified green's function

(See attached file for full problem description)

Frobenius norm of a matrix

(See attached file for full problem description)