Real Analysis Answers

darboux theorem

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

If a>0 and b >0 Show that

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

Fourier series for the function x ^ 2 in the period (-π, π)

Using the Average Rate of Change of a Function formula, how do I solve g(x)=1/x; x=1, x=a?

(x)= x^(2/3)

give an epsilon delta proof that f(x) is continuous at any c in R. Is f(x) uniformly continuous on R?

Give an epsilon-delta proof that f(x) is not differential at x=0

Given the function:

f(x1,x2) = 2/(x1x2) + x1x2 + x1,

Show that there is no solution of minimizing this function over x1 > 0, x2 > 0. Can f(x1,x2) ever be smaller than 2sqrt2?

Let f(x,y): R^2-->R, and for each fixed y, let inf(x)f(x,y) denote the greatest lower bound of the set of numbers {f(x,y)|x element of R}. Show that

infx(inf(y)f(x,y)) = infy(inf(x)f(x,y)).

noting that inf(y)f(x,y) <= f(x,y) for all x and y.

Let S and T be non-empty subsets of the real line with s <= t for every s in S and t in T. Show that sup(S) <= inf(T).

We want to minimize the function

f(x) = x^2 + 1/x^2 + 4x + 4/x.