Recent content by sandra1

Homework Statement If a function f is represented by the power series ∑(k=0 to ∞) a_k(x-a)^k, with a radius of convergence )<R<∞, then f is continuous on the interval (a-R, a+R) Homework Equations The Attempt at a Solution I don't know if my proof is loose at some point or not...

Thank you. So here's my trying. Would you mind looking through if there's any problem with it? I also have problem with the -R in part b. a. For R in (0, ∞), choose x in [a-R+ε1, a+R] with an ε1>0 so (x-a) <=R. Therefore: |(a_k)(x-a)^k| <= |(a_k)(R^k)| But since ∑(k=0 to ∞) (a_k)*R^k...

Hi everyone, I'm reading my analysis textbook and trying to prove Abel's Theorem but I don't really get it. Any help would be very much appreciated. Thank you very much! Homework Statement Suppose that R in (0, ∞) is the radius convergence for the power series ∑(k=0 to inf)...

Homework Statement f: [-a,a] >. R is Riemann integrable, prove that ∫[-a, a] ƒ (x) dx = 0 Homework Equations The Attempt at a Solution This only proof below I can think of is rather very calculus-ish.I wonder is there any other proof that is more Real Analysis level for this problem? Thanks...

yes you're right. i totally forgot about the general case with Q refines P by any finite number of points more than 1. Thanks very much for your response.

Homework Statement Suppose that function f: [a,b] --> R is bounded, and P and Q be 2 partition of [a,b]. Prove that if P is in Q then U(Q,f)<= U(P,f)Homework Equations The Attempt at a Solution P is in Q so suppose there's a c that is in Q but not in P such that c is in between x_i-1 and x_i...

thanks very much for your help.

thanks very much both of you for the hint. you're right i need a function with both increasing and decreasing parts. So ok from your hint f(x) = x-x^3. x is in [-1,1] So the real value of riemann integral is 0. pick P = {-1,0,1} Q = {-1,0,1/2,1} left-handed sum using P =...

you're right. i was wrong. so it must be that L(P,f) = 2/3 and, L(Q,f) = 17/24 so now L(Q,f) is closer so an increasing function wouldn't work. So a decreasing function must work right? so how about f(x) = -2x using the same two partitions P,Q then L(P,f) = 0 + (1/3)(-2/3) + (1/3)(-4/3) =...

Homework Statement Give an example of a function, interval, and partition P for which a left-handed sum using P is closer to the actual value of the Riemann Integral than the left-handed sum using partition Q which is a refinement of P Homework Equations The Attempt at a Solution...

Hi, oh yes, your function works. All functions f(x) = mx + n are bijections. Thanks for your help. About the second one, do you think I should break it into 3 cases. With a,b < 0; a,b > 0; and a <0 ^ b>0?

Homework Statement I really don't understand the question for this problem, could you please help me out? Thanks so much 1.a,b are some real numbers. Give an example of a bijection from [0,1] to [a,b] 2. Prove that all functions from [0,1] to [a,b] are bijectionsHomework Equations The Attempt...