Consider Hhe seguen ce (f,) of funcdions fa:R-R defined by: fn (x) = Cos for 番

Q: Use the series to evaluate the limit tan-¹ y - siny lim y→0 y³ cosy

A: Expand series and solve

Q: 1 (x + 1)2 dx -2

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Q: Example: Examine the convergence of COs x dx (ii) / x'e*² dx

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Q: 1 52. n2 + sin n n=1

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Q: Conclude on the convergence or divergence of sin x dx cos x

A: Since function is not continous at x=pi/2. Hence we break the limit of integral at x=pi/2. This…

Q: WNII sin 2 7) Show that 4 VN converges/diverges.

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Q: Detine s F: [o,0)IR, FCH=2 -tx sinx dx Use Dominated Convergence Theoren and show that: dF -1 eand…

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Q: Establish the convergence or divergence of the given sequenoes in full detail: (a) In = n!. (b) %. =…

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Q: Is Uniform convergence of {fn(x)} is sufficient but not necessary to transmit continuity from the…

A: From the theorem stated in step two, we can see that the function is uniformly continuous and we can…

Q: Use the definition of convergence to prove lim {n→∞} (cos n)/(n2-n+1) = 0.

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Q: (-1)"n" sin" (4) 2n n=1 CS Samed with Camacarner

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Q: • If it converges, what is its limit? Provide some justification for your answer. • If it diverges,…

A: key points By monotone convergence theorem ;  if a sequence is decreasing and is bounded below by an…

Q: 1 n4 + sin“ n=1

A: To find that the given deries diverges or converges

Q: 1. brove 1 prove that the inter val of convergence of the that the inter ual 2an +1 (nti)Sn or seri…

A: According to our company's guidelines I can only answer first question since you have asked multiple…

Q: cos“ n Σ 48. n2 n=1

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Q: If Cn(x – 3)" converges at z = 7 and diverges at z = 12, what can you say about: (a) the convergence…

A: Given that ΣCn(x-3)n converges at x=7 and diverges at x=12. We know that, if the power series is…

Q: Prove that sina de converges but not absolutely for 0 < p < 1. XP

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Q: Determine the convergence of the integral tan r UTMUT dr. E (In(cos r))2

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Q: 00 6n–1 Зп+2 1 Apply the nth term divergence test to conclude if E converges or diverges.

A: The divergence of a series can be found using the nth term test. Consider the series . Let . If…

Q: Determine the convergence of the integral 2 tan x UTM UTM dx. (In(cos x) GUTM

A: Given: ∫π3π2tanxlncosx2dx

Q: The radius of convergence of the power sorics n+1 2" is

A: here in this question given that power series and we find its radius. this is form of ∑an zn   and…

Q: By using the substitution r = t2 determine the convergence of %3D d.x ro (x+ 1)VT

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Q: Consider the sequence of functions n+ cos(x) 3n + sin(x) defined on the R. 1. Find the function f to…

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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: Determine the convergence of the integral tan r dx J; (In(cos x))²

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Q: In(1+3y) dx. Utilizando desarrollos limitados, determine la convergencia de la integral cos(x)-1-1

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Q: a] If the function f(x) is continuous in a closed interval [a,b] and f(a)f(b)<0, Prove that the…

A: let us prove the first part by considering a function let fx=x, x not equal to zero  and f0=1 but as…

Q: Using the nth term test for divergence, the series 00 08 An > V42n+1 %3D n=1 is divergent since lim…

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Q: 2. Show that following sequence of functions are convergence pointwise to its limit function. a)…

A: See the attachment

Q: 2. Let 3nr? + cos nr In (r): %3D (a) Find the pointwise limit of (gn) on R. (b) Is the convergence…

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

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Q: What does the nth-Term Test for Divergence say about the following scrics? use the Integral Test to…

A: if integral is finite then series is convergent

Q: Determine the convergence of the integral tan x d.x. E (In(cos r))2

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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: &= Let P and q ate palynamiels. and has no Yeal zetos. find necessaty and Sufficient conditions for…

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Q: Find the interval of convergence of f(x) dx where f(x) = (-1)+1(x-6)" 8. n%3D1 n6h

A: Given that             f(x)=∑n=1∞ (-1)n+1(x-6)nn 6n ⋯⋯⋯⋯ (1) To find the interval  of convergence by…

Q: Use an appropriate Riemann sum to evaluate the limit

A: Solution:- We have to use an appropriate Riemann sum to evaluate the limit…

Q: Determine the convergence of the integral tan x J, (In(cos r))² dx. UTM UTN

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Q: If E, 2"an converges and E o(-1)"2"an diverges, then the radius of conver- gence of o anx" is 2.

A: According to the given information, Suppose,

Q: b) Use the Comparison Theorem with f(x) = - to show whether , g(x)dx is convergent or divergent.

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Q: Determine the convergence of the integral tan x UTM &UT dx. (In(cos x))² COS X

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Q: 04: Determine the convergence of tan¹(x) x²+n²In(n+2) , on A = [0, +00).

A: We have to determine the convergence of the given series of functions ;   ∑n=1∞an = ∑n=1∞tan-1xx2+n2…

Q: Try to prove the convergence of Newton’s Method

A: The given problem is to prove the convergence of the Newton's method, We have to derive the…

Q: Suppose fn : [0, 1] → R be defined as f,(x) = xn , for n e Z+, for x E [0, 1]. Define: f(x) := { 0,…

A: Given that fn(x)=xn (a) Define, f(x)=0x∈[0,1)1x=1 Take fn(x)=1xn. We say that fn→f pointwise if…

Q: Q No 2: a) Discuss the convergence and divergence of (1+ a"), Va eR.

A: We have to divide it into several cases.

Q: fn (x) = Vx e (1, 00), Vn E N. Explain why the sequence {fn} does not satisfy the conclusion of the…

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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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Q: Define s F: [0,0)-IR, Fa- esinr dy Use Dominated Convergence Theoren ond show that: dF (H= at -1…

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