Let the sequence {fn(x)} be defined on [0, 1] by n Рn2x; if x E (0, ) fn(x) : otherwise SHOW THAT: 1 1 limno | fn(x)dx | limn°fn(x)dx.

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

A: Given - x1>1 and xn+1=2-1xn for n≥2 To prove - xn is convergent and also find its limit .

Q: Suppose lim, An = 1. Prove that 1 lim E(-1)*a2k = 0. n-00 n k=1

A: Given that limn→∞an=1. To prove limn→∞1n∑k=1n-1ka2k=0.

Q: Let n be a positive integer. For which n are the two infinite one-sided limits limx→0±1/xn^n equal?

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Q: Note: You may need to assume the fact that  lim M→+∞ Mne−M = 0  for all n.Sales of the text Calculus…

A: To calculate the total future sale of the text.

Q: 3. Let f : R - {0} → R be such that the sequence (xnf (xn)) converges to 3 whenever (xn) is a…

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Q: 00 Fn+1 { is convergent, and Show that the sequence Fn n=1 1 Fn+1 lim n→∞ Fn Fn+1 > 1 and Fn-1…

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Q: = 3. Find the pointwise limit g of the sequence {n} on I = [0, 0), where gn(x) = the E-definition…

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Q: Find a formula for the sum of n terms. Use the formula to find the limit as n → ∞. in (음)(금) 6i lim…

A: limn→∞∑i=1n6i x 2n2First we find sumation∴∑i=1n12in212n2∑i=1ni…

Q: Show that if Σ i=1 converges towards a € R , then lim ) a; = 0 i=N

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

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

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Q: Express a = the limit lim Σ(4(x*)² – 3(x* )³) Aï; over [6, 7] as an integral. n→∞ i=1 Provide a, b…

A: The given limit limn→∞∑i=1∞4xi*2-3xi*3∆xi over the interval 6,7

Q: -3 sin n Consider the sequence b, = . In order to apply the squeeze theorem and determine the limit…

A: To find the limit of bn-3sin2nn using squeeze theorem as n goes to infinity.

Q: Use series to evaluate the limits

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Q: limit as n approaches +infinity of (n/(e)^n)

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Q: Theorem 1 states that if lim x→∞ f (x) = L, then the sequence an = f (n) converges and lim n→∞ an =…

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Q: Use series to evaluate the limit. sin(3x) – 3x + x3 lim -

A: Given,        limx→0sin(3x)-3x+92x3x5=

Q: Apply the ratio test to the series. n! 2n3 n=1 an+1 Find the limit lim an

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Q: 1. Show using sequences of functions that lim -nz? e dx 0.

A: Solution is below

Q: Define the sequence of functions {Jn(x)} on [0, 1] by 2n?x; 2n(1 — пх); 0; 2n fn(x) = 2n くE Find…

A: The given sequence: fn(x)=2n2x;                    x≤12n2n(1-nx);   12n<x≤1n0;…

Q: 1. Determine the infinite (oo or -o) limit as follows: x² - 2 (a) lim x2-x² - 3x + 2 sin x 1 x+ cos…

A: Topic = Limits

Q: tan-'(x) – In(1 +x) Evaluate the limit lim using series. x-0 x2

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

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Q: 4) Make a table for a sequence of carefully chosen values of a to estimate the limit 1® – 1 f(x) =…

A: By using table value of x for this given function to estimate this given limit

Q: Use series to evaluate the limit. 4 3 sin(2x) – 2x + lim 4/3

A: Given problem:-

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

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Q: a Exponential limit Prove that lim (1 + e", for a + 0. %3D

A: Given limx→∞1+axx=ea, a≠0

Q: 3. Show that if x > 0 for all n e N, then lim(x,) = 0 if and only if lim(1/x,) = +oo.

A: Consider the given information. xn>0 and limxn=0 Now, use the limit definition of the sequence.…

Q: Use the Squeeze Theorem to find the limit of the sequence. sin() 4n sin

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Q: Give an example of a sequence of continuous functions: fn: [0, 1] R that is pointwise convergent but…

A: Consider the function fx=0, x is irrational in 0,1mn, x is rational in 0,1. This function is called…

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

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Q: Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].…

A: According our guideline i solve only first.

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: 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: Let an 2n+4n-3 Using the definition of the limit, prove that lim,∞ an 3n2+2n+1* 2/3.

A: Given that, the rational function  an = 2n2+4n-33n2+2n+1 To prove that limit of function is limn→∞…

Q: 3. Let f : R- {0} → R be such that the sequence (xnf(xn)) converges to 3 whenever (n) is a sequence…

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Q: 10. Find the limit (if it exists) of the sequence (x_n) where x_n= cos(n)/n.

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: Show using sequences of functions that 2 lim e dx = 0.

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Q: 4. Find the limit, if it exists or show the limit does not exist: a) b) lim 47,² (x,y)(0,0) x + y…

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Q: Use series to evaluate the limit. 2 +2 cos t-2 lim t+0

A: Given,           limt→0t2+2cost-2t4

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

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Q: 1. Let f: (0, 1) R be uniformly continuous. Let (r) be a sequence in (0, 1). Suppose that lim, x In-…

A: We will show lim f(xn) exists. For this we will show f(xn) is a Cauchy sequence of real numbers.

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

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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: Use series to evaluate the limit. 1- cos(5x) lim x → 0 1 + 5x – e5x

A: Authoring guidelines: As per the authoring guidelines we can solve only the first question.Please…

Q: Define the sequence of functions {fn(x)} on [0, 1] by { 2n2x; 2n(1 — пх); т. fn(x) : 0; n Find f(x)…

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

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Q: 2. If (xn) and (Yn) are sequences of real numbers such that lim (xn) + 0 and lim (xn Yn) exists,…

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

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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.