Real Analysis Answers

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}?

Assume a = a(t) ∈ L1(0,1)∩C(0,∞), ∞ ≥ a(0+) > a(∞) ≥ 0 and a(t) ≥ 0, a′(t) ≤ 0, a′′(t) ≥ 0, a′′′(t) ≤ 0, on (0,∞). Let h(t) = ta(t) on (0,∞). Then one of the following is true. There exists a number ε > 0 such that h is increasing on (0, ε). Or, there is no such interval. Which is correct? Prove it.

darboux theorem

For which real numbers x does the series ∑_(n=1) to n = infinity (√(n+1)-√n)/n^x converges?

let s be the subset of real number then if any natural number u having the properties 1-u-1/n is not an upper bound and 2- u+1/n is an upper bound then u =Sup(S).

If a>0 and b >0 Show that

lim n->infinity squareroot((n+a)(n+b)) -n = (a+b)/2