Recent content by applegatecz

Homework Statement Find all values of p for which the given series converges absolutely: \sum from k=2 to infinity of [1/((logk)^p)]. Homework Equations The Attempt at a Solution I've tried the ratio test, the root test, limit comparison test ... everything. I know the answer is...

There is no fraction in the original summation.

OK, I think I understand: the expression factors to x^(k-1)*[x^(k-1)-x^(k-1)] = 0?

Hmm, yes, I didn't think of that. But in that case, I get that the value of the entire integral approaches infinity as x-->infinity: all the terms go to zero except that second term [(x^2)/2!] / x^2, which is 1/2; the antiderivative is therefore x/2, which approaches infinity as x-> infinity?

Hmm -- what do you mean? I did graph the function, and it seems apparent that it is indeed improperly integrable -- but I don't think stating as much is a sufficient response.

Ah, I see, thank you. In this case, unlike the "regular" geometric series case, x can be less than one OR one (i.e., |x|<=1), because if x is one, x^2k and x^(2k-1) are both one ... so the series converges to zero. Correct?

Homework Statement Decide whether f(x)=\int (1-cos(x))/x^2 is improperly integrable on (0, infinity). Homework Equations The Attempt at a Solution I understand the concept of improper integration, but I don't see how to take the antiderivative -- I tried substitution and by parts...

But then x^2 is one of the factors, and x^2 does not converge (?).

Homework Statement Find all x for which \sum from k=1 to infinity (x^k - x^(k-1))(x^k+x^(k-1)) converges. Homework Equations I think the geometric series formula is relevant: \sum k=N to infinity of x^k = 1/(1-x) for all |x|<1. The Attempt at a Solution I simplified the expression...

Ah, I see! Thank you.

Homework Statement Show that ln(e)=1. Homework Equations ln(x)=antiderivative from 1 to x of dt/t The Attempt at a Solution I assume we have to use the fact that e= lim as n->infinity of (1+1/n)^n, and perhaps can apply l'Hopital's rule to transform that limit -- but I'm not sure...

Homework Statement I would like to factor (x^n-a^n) such that (x^2-a^2) is one of the factors. Is this possible? Homework Equations The Attempt at a Solution I tried to get this with a kind of reverse polynomial long division, but couldn't do it.

Oh! Somehow I missed that. Thank you very much for your help.

Homework Statement Prove that x^n approaches a^n as x approaches a. Homework Equations The Attempt at a Solution I understand the concept here ... need to find a delta>0 for epsilon>0 s.t. |x-a|<delta implies |x^n-a^n|<epsilon. For some reason I can't solve this one. Thanks.