Q(A). Let {fn(x)} defined over [2,3] Show that (a) fn(x) is meaurable and monotonic increasing for all n. (b) (Sn(x)} converges a e to a function f(x) to be determined be a sequence of functions 1+(x– 2)* J. (c) Apply the monotone convergence theorem to evaluate Lim In(z)dµ.

Q: x2n-1 Find the radius of convergence of En=1 5n 100

A: Given, ∑n=1∞ x2n-15nFind radius of convergence

Q: Let x1 = 1 and xn+1 1 for n e N. Show that this sequence is convergent and find its In + 2 limit.

A: I have shown the sequence is bounded above and monotone increasing which implies sequence is…

Q: (ii) Prove that if a sequence Xn converges to a limit I, then any subsequence of Xnalso converges to…

A:

Q: (en) is ultimately decreasing and that x:= for lim(xn) exists. h Use the fact that the subsequence…

A: Given: xnis a bounded sequence, sn=supxk:k≥n and sn=infsn

Q: Let fn(x) be the following sequence (not series) of functions n2 fn(x) : (x2 + 7x + 6)n² +n (1) Is…

A:

Q: (B) Let {fn(x)}ı = {xlnx. cos(x –1)"}1 be a sequence of meaurable functions defined over [1,2]. Show…

A:

Q: Suppose that the sequence {n} is such that xn 20 for all n eN and that lim(-1)"xn exists. Show that…

A: We will use Triangle inequality.

Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A: Proof

Q: Consider the decreasing sequence of functions {fn} defined on R, where fn(r) = X[n,00), Vn e N. That…

A:

Q: 1. shown that I and I norms on R" are equivalent with respect to convergence and show that the…

A: It is given that,  l2 and l∞ are the equivalent norms on ℝn with respect to convergence. We have to…

Q: Question 6. In Example 2.4.3, show that f(r) converges to 0 at each r E [0, 1].

A: Let us consider the function :                                  fnx=n if 0<x⩽1n0 otherwise We…

Q: 1.1 Let {fn (2)} = {1 NI +nz ,z ER and 2 > 0. 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A: To Examine: The series of function Sn(x)=∑k=1∞xk-1 converges uniformly on [-a,a], a∈(0,1) or not.

Q: V. Let f(x) be a differentiable function such that for all x > 1, f(x) • f(x) > 1, f'(x) (-1)" f(n)…

A: Solution is given below:

Q: 1.1 Let {fn(x)} = { ,x ER and r > 0. %3D + xn 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A:

Q: 1.1. For r € (0, 1] (1) = f(r), wher

A:

Q: Let {fn(x)} be defined by: fn(x) = Vx for 0 0 you can find a point 0 < x < 1 such that | VI – 1| 2.…

A: Given fn(x) is defined by, fn(x)=xn, x∈0,1 (a) Let ε&gt;0 and let x∈0,1 If x=0, fn(0)=0  So, take…

Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim xn = inf {xn…

A: Multipart: (A)  We need to prove that , if xn is decreasing bounded sequence then , xn is convergent…

Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n²x; if x E (0, ) = (x)"f 0; otherwise Find the…

A: we have sequence of function fnx=n-n2x ; if x∈0,1n0;           otherwise pointwise limit of sequence…

Q: 1. Let {gn} be a sequence of integrable functions which converges a.e. to an integrable function g.…

A: Solution: Given gn is a sequence of integrable functions which converges a.e. to g. Also, fn is a…

Q: 3. Let f(x) = x³ – 3x – 2 (a) Define the Newton-Raphson sequence {Xn} for the given f(x) (b) Start…

A: Given a function,

Q: 4. Consider the folowing sequence of functions. 5+ nx? fn(x) = , x E [0, 1] 5+ nx a) Find the…

A: (4.) We will find the all step of the function.

Q: 1-e Determine the rate of convergence of the function F(h) = as h 0. O A. O(h^3) O B. O(h) OC.0(1/h)…

A:

Q: Consider the sequence of functions {fn}nen, where nx fn(z) = Va € [1, 10]. 1+ nx' (a) Find the…

A:

Q: 4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn…

A:

Q: find if diverges or converges. if converges, then find its limit cos²(8n) An 8n-1

A: Given - an = cos28n8n - 1 To find - find if diverges or converges. If converges, then find its…

Q: Study uniform convergence on 10, 1) of the sequence of functions Jn for the following: a fn(x) =…

A: Solution a: Given that fn(x)=11+(nx+1)3 in 0,1. Now, f(x)=limn→∞fn(x)=limn→∞11+(nx+1)3=0,    x≠012,…

Q: (b) Let X and explain whether the sequence X = (Xn) converges or diverges. (xn) be defined by xn =…

A: We have to solve 4.b

Q: V. Let f(x) be a differentiable function such that for all æ > 1, • f(x) > 1, • f'(x) < f(x) and lim…

A:

Q: Let {fn}=1 be a sequence of continous functions on an interval [a, b]. If {fn}n=1 converges…

A: As per the the instruction in the question, we will be solving part (a) - (i). Let fnn=1∞ be a…

Q: I have no idea how to approach this question

A: Known fact:

Q: (iii) 2 an must converge if lim an 0. n =0 bn (iv) 2 bn must diverge if b, > 0 and lim = 4. n->0…

A: The solution are next step is

Q: nx Let {fn(2)} = {1 for x ≥ 0. 1+nx . (a) Find lim fn(x) = f(x). n-x (b) Show that if t > 0, the…

A:

Q: 1.2 Find the pointwise limit of the sequence of functions {fn(2)} = {nc(1 −z)} on [0, 1] and…

A:

Q: Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 4]. 4…

A:

Q: Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for…

A: Given: x1=1, xk+1=2xk So, x2=2x1=2&gt;x1 x3=2x2=22&gt;x2 On generalizing, we get…

Q: 1.1 Let {fn(#}} = {1 ng „I ER and r > 0. 1+ nx 1.1.1 Find the pointwise limit of the sequence {fn…

A:

Q: Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 4].…

A: Given limit is                        limn→∞4n4n+8n+12n+...+4nn Again…

Q: Let {fn(x)} x E R and x > 0. 1+ xn use defination of unifrom convergence to Show that the…

A:

Q: Consider the sequence of functions {fn}nen, where nx fn(2)= Væ€ [1, 10]. 1+nx (a) Find the pointwise…

A: Given :  fn is a sequence defined by ;  fn(x) = nx1+nx ,  ∀ x∈[1, 10]. To find :  (a) The pointwise…

Q: Q(A). Let {fn(x)}=1 = 1+ (x – 2)* , be a sequence of functions | n=1 defined over [2,3]. Show that:…

A:

Q: Let {fn}%=1 be a sequence of continous functions on an interval [a, b]. If {fn}-1 converges…

A: Given: (a) (ii) To show that the sequence fn is not uniformly convergent on x∈0, 1, where…

Q: 8. Prove that lim(x_) = 0 if and only if lim(jx,i) = 0. Give an example to show that the conver-…

A:

Q: 11. Let a, b ER with a < b, and let {fr} be a uniformity convergent sequence of continuous real-…

A:

Q: 2. Let gn(x) to zero uniformly on [a, o). Show that the convergence is not uniform on [0, 00). for n…

A: Given function is gn(x)=nx1+(1+n2x2) Now limn→∞gn(x)=g(x)=limn→∞nx(1+n2x2)…

Q: 3. Let h„(x) : zero uniformly on [0, b]. Show that the convergence is not uniform on [0, 1] for n E…

A:

Q: Let {fn}1 be a sequence of continous functions on an interval [a, b]. If {fn}-1 converges uniformly…

A: Given that fnn=1∞ is s sequence of continuous functions on an interval a, b.We have to prove or…

Q: Q(A). Let {fn(x)} = be a sequence of functions 1+ (x – 2)" defined over [2,3]. Show that: (a) fn(x)…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: find if diverges or converges. if converges, then find its limit cos² (8n) an where n 2 0 8n-1

A:

Q: 1.1 Let {fn(2)} = {1 nr ,z ER and 2 > 0. 1.1.1 Find the pointwise limit of the sequence {fn(r)} if…

A: The given sequence of functions is fn(x)=nx1+nx, x∈ℝ and x≥0. 1.1.1. To find: Pointwise limit of…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.