Real Analysis Answers

Test the following series for convergence:
a) ∑(n goes from 1 to infinity) n^(1/n).
b)∑(n goes from 1 to infinity) (((-1)^n)sin(3n x))/n^3

Show that: ∑(n goes from 0 to ∞)1/((α+n)(α+n+1)) =1/α , for α>0.

What is the derivative of a function f(x) at a point c given by?

What does Rolle's Theorem state?

Suppose that S is contained in R and that S is not closed. Is it true that there is a subsequence in S that converges to some x not in S?

Prove [0,1] is not countable without using the outer measure of an interval is its length?

Determine if the series is absolutely convergent, conditionally convergent or divergent
Σfrom m=1 to infinity {sin([(1+2m)π]/2) / (m+1))ln(m+1)}. Thanks in advanced.

Find the supremum and infimum of the set {(-1)^n(1+1/n):n is natural

Let S⊆R be non empty . Prove that if a number u in R has the properties (i) for every n∈N the number u-1/n is not an upper bound of S, and (ii) for every number n∈N the number u + 1/n is upper bound S, then u =sup S.

Is the function f:R^2-{(0,0}→R^2 defined by
f(x,y)=(x/(x^2+y^2 ),-y/(x^2+y^2 ))
continuous on R^2-{(0,0}?