Recent content by ohreally1234

Homework Statement Suppose sum(a_n*x^n) represents a power series with radius of convergence (-R, R). Is it true that the series sum(n*a_n*x^n) is convergent? Prove or give a counter example. Homework Equations The Attempt at a Solution Let b_n = n*a_n*x^n Using ratio test...

ooo that makes so much more sense: range of f is [0,2] but the inverse f^-1: [0,2] => [0,1) U [2,3] is clearly discontinuous at x=1. thanks so much

sorry I am stumped... i tried playing with functions such as f(x)=1/(x-1) but I am not sure what to do

If the domain is non-compact, does such a function exist?

I'm having trouble with giving a counter example of a continuous 1-1 function f:[a,b] => R whose inverse is not continuous (does it even exist)?.

oh sorry, i also forgot to add that f:[a,b] => R I got the continuity part down (your hint really helped!), but I'm having trouble with the compact part.

But isn't f(x) = (cos(x), sin(x)) not a 1-1 function (because it's a circle)?

Hmm, how do I prove the first part (that it is continuous)

Homework Statement Prove that the a continuous function with compact domain has a continuous inverse. Also prove that the result does not hold if the domain is not compact. Homework Equations The Attempt at a Solution I tried using the epsilon delta definition of continuity but...

hmm yeah i thought that part of my argument was a bit shady. can anyone offer some insights?

I reasoned that M exists because the real numbers are dense. and you can prove M^(1/x) goes to 1 using the definition of the limit. Are there any holes in my argument?

sorry i meant that lim goes to 1. (fixed the typo in original post)

Homework Statement Consider the sequence a_n = abs(sin(x))^(1/x) Find the lim a_n if it existsHomework Equations None. This is for my calc 2 class. The Attempt at a Solution We are studying the sandwich theorem, so I thought 0 < M^(1/x) < abs(sin(x))^(1/x) < 1^(1/x). (Because I assumed that...