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
Give a counter example to negate the statement that for functions f_1, f_2 :[a, infinity) --> R with c_1 and c_2 are real constant, we have:
∫[a,∞] (c_1)(f_1) +(c_2)(f_2) dx = c_1 ∫[a,∞]f_1 dx + c_2 ∫[a,∞] f_2 dx
Homework Equations
The Attempt at a Solution...
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 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...