Homework Statement
For what p is f(x)=log^a(x)/x^p integrable from (1,infinity)?
Homework Equations
The Attempt at a Solution
I'm not sure, maybe I can do some sort of inequality so that I can arrive at the comparison theorem for improper integrals.
Homework Statement
Suppose that a_n->L as n->infinity. Show that (a1+a2+...+an)/n=L as well.
Homework Equations
The Attempt at a Solution
I'm thinking something about limit theorems here?
Homework Statement
Let x_1=1 and let x_n=x_n-1 + 1/n^n for n>1. Show that x1, x2, ... is convergent.
Homework Equations
The Attempt at a Solution
I have managed to transform x_n=summation(1/n^n). How do I show that this is convergent?
Thanks for the explanation. The problem is clearer now to me. What I still do not get is how the epsilon delta definition guarantees it. ?? Sorry if I can't follow through... I'm new to this.
Homework Statement
If f is a real function which is continuous at a element R and if f(a)<M for some M element of R, prove that there is an open interval I containing a such that f(x)<M for all x element of I.
Homework Equations
Extreme value theorem, intermediate value theorem...