3. Define the functions gn (2) TL n+1 0 OO 2-¹
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![3. Define the functions ga(z) -
{
TL
n+1
0
2-1 << 2
else
Then consider the function () = 9(r) defined on [0,1].
78=1
on the set [0, 1].
12
(a) Using the partition P= {0,; 1} find the upper and lower Darboux sums for f.
(b) Prove or disprove that is integrable on the interval [0, 1]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd90f20f1-69b1-40a4-b599-dc62599940c2%2F04ab4932-ec36-4685-a292-1e4716408a89%2Fptjhnzv_processed.jpeg&w=3840&q=75)
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- For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.If u(z) is an analytic function in the unit disk and has the Taylor expansion \Sum_{k=0}^{\infty}b_k z^k, then prove that \Sum_{k=0}^{\infty}\dfrac{|b_k|^2}{k+1} converges.(a) Prove that a bounded function f is integrable on [a, b] if and only if there exists a sequence of partitions (Pn)∞n=1 satisfying
- 1 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.Prove that a polynomial of degree n is uniformly continuous on R if and only if n = 0 or n = 1.Suppose that K is a Riemann integrable function on [0.3, 31.5] and 7(9) = K(9)except for the values of 9 ∈ [0.3, 31.5] ∩ ℕ.a) Enumerate all the values of 9 in which 7(9) ≠ K(9). b) Is 7 also a Riemann integrable function on [0.3, 31.5]? Why?
- A function f : N × N → N is defined by f (m, n) = 2ᵐ−¹(2n − 1).(a) Prove that f is one to one and onto.(b) Show that N × N is denumerable.b) If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an bn) are positive?Let A be a non-empty and bounded subset of R, and let x_0=supA. Prove that x_0 ∈ A or that x_0 is an accumulation pt of A.
- determine the following values for the nth partial sumof the Fourier Series of the given function:fx=×+4.I-7, 7]Let f be a bounded function on [a,b]. Prove that f is integrable on [a,b] if and only if there is a sequence of partitions {Pn} of the interval [a,b] such that limn→∞ (U (f,Pn)) - L(f,Pn) = 0Prove that f(n) = 1000n5 + 20000n2 + 32 is O(n6 ); Is this a tight upper bound? Why or why not?