Recent content by Rosey24

Homework Statement We have a worksheet with practice final questions and I'm really stuck on this one on continuity: Suppose h: (0,1) -> R has the property that for all x in (0,1), there exists a delta>0 such that for all y in (x, x+delta)\bigcap(0,1), h(x) <= h(y) a) prove that if h...

I am allowed to use uniform convergence. Thanks!

Homework Statement Show that \Sigma (from n=1 to infinity) of sin(nx)/n^2 is continuous on R Homework Equations The Attempt at a Solution No idea, any help would be greatly appreciated.

so would taking f(x) = x^3, which is continuous, be a suitable counterexample for the first assumption? I can't think of a function that is always positive and isn't continuous, though.

Homework Statement The original question required me to show that for f(x) >= 0 for all x, f continuous, where the integral (from a to b) of f =0, that f(x) = 0 for all x in [a,b]. I did that, using a proof by contradiction. Second part of the question requires me to show that the two...

Homework Statement I need to use Taylor's thm to get an approximation to sqrt(5) with an error of no more than 2^(-9) and am totally lost. Homework Equations Taylor's theorem: Rn(x) = f(n)(y)/n! *x^n -- where f(n) is the nth derivative of f and Rn is R sub n. The Attempt at a...

Homework Statement We recently proved that if a function, f, is continuous, it's absolute value |f| is also continuous. I know, intuitively, that the reverse is not true, but I'm unable to come up with an example showing that, |f| is continuous, b f is not. Any examples or suggestions would...

as n goes to infinity, (n+1)n will converge to 1, right? and log (1) is zero.

I recalled this incorrectly, it's log[(n+1)/n] and no bounds were given, though I would assume it's 1 to infinity as zero would diverge.

Homework Statement I'm supposed to evaluate the following series or show if it diverges: SUM (sigma) log [(x+1)/x] Homework Equations Drawing a blank...:confused: The Attempt at a Solution I'm unsure how to start this. We've gone over all sorts of tests for convergence (ratio, comparison...

Thanks to everyone who responded! I appreciate the simplicity of your response, huyen_vyvy. One small note: it's a she who assumed r=1, not a he.

I need to prove the following: n!/r! >= r^(n-r) With r and n as natural numbers and n>=r I know the LHS will end up being (n-r) terms long as the first r! will cancel out of n! (n>=r), but as they're both unknown, I just left it as 1*2*3*...*(n-2)*(n-1)*n 1*2*3*...*(r-2)*(r-1)*r and I looked...