Recent content by pcvt

Do you mean I can use that to get a taylor series for my function

I don't see an easy formula for the nth derivative so I'm not sure how to proceed in that direction

I've seen this thread: https://www.physicsforums.com/showthread.php?t=297842 and that is the exact question I need to to answer. What is the radius of convergence of 1/(1+x^2) expanded about x_0=1? The problem is, I can only use an argument in real analysis. I see the answer is...

From there I get down to finding for what values the limsup of 1/n x^(2^n/n) is less than one for I believe its |x|<=1 but I'm not sure how to analytically prove this

Homework Statement Find the radius of convergence of sum from 1 to n of 1/(n^n) * x^(2^n) Homework Equations The Attempt at a Solution Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent

Meant to say I have it down to proving that a strictly increasing, onto function: f: R->Q is continuous

Homework Statement Prove that there is no strictly increasing function f: Q->R such that f(Q)=R. (Do not use a simple cardinality argument) Homework Equations The section involves montone functions, continuity and inverses. I believe the theorem to be used is that a monotone function on...

So helpful, got it from there. Thanks!

Ok, I see that I can show it for f(1/n). How can I combine the two though, since I don't have a rule to say f(m/n)=f(1)*(m/n)? Also, how can I include negative numbers? Thanks!

Homework Statement Suppose that a function f R->R has the property that f(u+v) = f(u)+f(v). Prove that f(x)=f(1)x for all rational x. Then, show that if f(x) is continuous that f(x)=f(1)x for all real x. The Attempt at a Solution I've proved that f(x)=f(1)x for all natural x by breaking up...

Well, would it be possible to redefine a new variable so that one can prove n(n-1)>0 for n>1 using induction? It seems possible to use induction but I'm not sure what to do about the fact that the statement isn't true for n=1.

Homework Statement State whether or not the sequence {a_n} = n+[(-1)^n]/n is monotone or not and justify. Homework Equations The Attempt at a Solution It clearly appears to be monotone increasing, so I attempt to prove this. I've tried using induction and tried to prove that...

Wouldnt that just say n^1/n converges to (n-1)^1/n which is trivial? I'm trying to manipulate the binomial formula, archimedean property, and bernoullis inequality to somehow get it but still haven't been able to after a couple hours

Hmm our class hasn't formally introduced logs yet so I don't know if it would be kosher or not to prove it using the log.

1. Prove that n^(1/n) converges to 1. 3. I've attempted to define {a} = n^1/n - 1 and have shown, using the binomial formula, that n=(1+a)^2>=1+[n(n-1)/2]*a^2. I think I'm on the right track but don't know how to bring this back to the original problem to prove convergence even after staring...