Let x1 > 1 and  xn+1 = 2 − 1 / xn for n ≥ 2.  Show that ( xn ) converges and find its limit.

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

A:

Q: Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E…

A: Given Information -  f is a continuous function on R and is 2π periodic.

Q: Find 2x dx. x2-4 (if it converges)

A:

Q: The sequence an =n-Vn +1vn+3 converges to the limit A)-3 B) -1 C) 1 D) +2 E) -2 F) 0

A: Given:

Q: Suppose x1 : 1/2 and xn+1 := x(n/2). Show that {xn} converges and find lim xn.

A:

Q: (d) Let x* = 1/2(2*), and what is the limit of x* as k 0? What is the order of converge for x?

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Use the definition to show that dx 2x2 converges.

A: We have to show that the given integration converges.…

Q: {VI2n + 8 – V12n} converges or diverges. If it converges, find the limit.

A:

Q: 1 -dx xln*x 1.

A: To determine if the following is a convergent or divergent. ∫e∞1xln4xdx Solution: A indefinite…

Q: Show that for every positive function f on R there is a monotone increasing sequence of integrable…

A: We need to show that for every positive function f on ℝ there is a monotoneincreasing sequence of…

Q: Suppose to solve for a root of f(x), i.e., ƒ(x*) = 0, we use the iteration Xk+1 = ¢(xk) where o(x)…

A:

Q: Find an example of fn → f in measure, but for every x, fn(x) does not converge to f(x).

A: We have to find an example of fn→f in measure, but for every x, fn(x) does not converge to f(x).

Q: 3^n+2 / 5^n converge or diverge?

A: f(n)=3n+25n=3n325n=9(35)n The sequence converges as 35<1.

Q: Evaluate (x2 – 2x³) dx using the limit of Riemann sums.

A: given integration is                                       I=∫01x2-2x3dx we have to find value of…

Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx a. Show that fn(x) converges pointwise to…

A:

Q: Given that lim n2(x-5)" converges by the Ratio test. If xEZ then the numerical 1- 00 value of x is…

A:

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

A:

Q: For each n ≥ 1, let fn : R → R, fn(x) =nx/(n + x^2) Show that the sequence of functions {fn}n≥1…

A:

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: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

Q: 2. s) Let x1 = 1 and define rn+1 = v3+2x, for n E N. Show that the sequence (xn) converges and find…

A:

Q: Suppose x1 : 1/2 and xn+1 := xn2. Show that {xn} converges and find lim xn.

A:

Q: 1 d.x x In (ex)

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: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

A:

Q: Let x1 O and xn+1 x+1 2 Show that (x,n) converges and find its limit.

A: It is given that  x1=0 and xn+1=xn2+12. Let xn+1=ϕ(xn)=xn2+12 By fixed point iteration method…

Q: 10. (x- 3)" Ln=1 n

A:

Q: DA. The sequence {(- 0.002)"} converges by oscillation to the limit |

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>x1 x3=2x2=22>x2 On generalizing, we get…

Q: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

A: A sequence of real-valued functions considers two types of convergence. These are pointwise…

Q: Find an example of a sequence of continuous functions {fn} defined on the interval [0, 1] such that…

A:

Q: Let g(x) = E-o (-1)" (2n): Find the interval of convergence g.

A: We have given g(x).  We have to find the interval of convergence.

Q: n sin"(n) :) has a convergent subsequence. Vn2+2n Show that (an) = (-

A: Consider the sequence an=nsinnnn2+2n We have to show that sequence an has a convergent subsequence.…

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn- 1+ Yn – 1) and y, = Vx, – 1 Ya-1 V n 2 2.…

A:

Q: State whether the sequence an = Vn +7 converges and, if it does, find the limit. a) O converges to…

A:

Q: Use series to evaluate the limit. V1+x-1 1 ーX lim

A:

Q: Given f: [2, )IR, prove that fn (x) = けx" It, converger uni formly to f6)-o.

A:

Q: Suppose f and g are continuous functions. Provide an example such that | g(x)dr diverges 0. adf(x)dr…

A: Follow next step

Q: that converges to a function on R ii)show n х 1=0n!

A: For this, we will use the Ratio Test, which can be written for a series ∑n=0∞an as: limn→∞ an+1 an…

Q: 4. Let (#1, y1) = (0,0) and let (n+1, Yn+1} = (2+ yn, 3 – Tn) for n > 1. Prove that (Cn, Yn) is a…

A:

Q: The sequence a=n-vn+1Vn+3 converges to the limit A) -3 B) -1 C) 1 D) +2 E) -2 F) 0

A: For a series ; if it converges then . If  the limit ; then the series diverges.

Q: Let f : [0, 1] → R be a continuous function. Show that the sequence of functions 1 fn (x) = ,x +1 +…

A:

Q: 7. Let x₁ = a > 0 and Xn+1 = x + 1/x,, for n E N. Determine whether (x,) converges or diverges.

A: Q7 asked and answered.

Q: Give an example of a sequence of continuous functions: fn : [0, 1] → R that has no pointwise…

A:

Q: 7. Suppose (fn) is a sequence of continuous functions on an interval I that converges uniformly on I…

A:

Q: Given that lim n²(x- 5)" converges by the Ratio test. If x e Z then the numerical n- 00 value of x…

A: Introduction: The ratio test is also used to test the convergence of a sequence. According to the…

Q: Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly…

A: Given, fx=nxe-nx , x∈0,∞ Since, limn→∞fx=0 , ∀x x=0, fx=0 and e-nx→0 as n→∞ Let, fx converges…

Q: Find the interval of convergence of Ln=1 (r-1)" 2n

A:

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

A:

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

A:

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.