Test the convergence and divergence of the following: 1 nin n 2. Do as required: 2. Σ=1 (n+3)! 3. En-2 cosnn n!37 3√√n 5. En=1 √n

Q: 1. Show that the series COS X 1 + cos 3x 3 is convergent whenever x is not an integer multiple of .…

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Q: Construct the inverse matrix (if it exists) for the following matrices: [12 61 (а) А %— 4 2. 1 -2]…

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Q: Q3. Use the Maclaurin expansion of er and the definitions sinh x = (e* – e-*)/2 and cosh x = (et +…

A: To show: a) sinhx=x+x33!+x55!+....                b) coshx=1+x22!+x44!+.... Concepts used: 1.…

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Q: Evaluate the following limit using Taylor series. 7 7 tan 1(x) – 7x+: lim 4x° 7 .3 7 tan (x) – 7x +…

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Q: cos n! + A) diverges B) converges 3628800

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Q: Let f = C(R/Z; C), and for n= 0,1,2,... define 2/₁₁ (2) cos(2x)da; by = 2(a) sin(2xnx)da. 0,1] Show…

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Q: (e) To what limit(s) can the fixed-point iteration In 5 cos(n-1)+2-1+kxn−1−4 1+k converge as n→∞,…

A: Given: xn=5 cos(xn-1)+xn-12+kxn-1-41+k

Q: 2.2.1. Prove that each of the following sequences converges to zero. write a) xn =…

A: We have to show that the following sequences  converge to zero.

Q: b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem.

A: As directed in the question, answer of only part b is provided below :   (b) Bolzano-Weierstrass…

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

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Q: Euler's Tution function: (n) is the number of positive integers less than n that are relatively…

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Q: 11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive…

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Q: 2. Consider the function f(x)=x²+5 on the interval [1,3]. If the interval [1,3] is divided into a…

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Q: cos? cos n 48. n2 n=1 IM:

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Q: Suppose that An +1 an →1 as n → ∞. Find the radius of convergence. n=1 an (1) xn 3n

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Q: 2. (a) Show that k!n-k-1 = for n>0 and ka nonnegative integer. 5. e-nxxk dx

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Q: z2n+1 7. Prove that sin z = (−1)”. converges absolutely for all z € C. Use (2n + 1)! n=0 this fact…

A: See below for detailed solution:Explanation: Step 2:Use the fact that the series for sin(z)…

Q: 7. Prove by induction. For any n2 0, E F = F,Fn+1• k=0

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Q: O which of the following Is true? a)E nCat2 is divengent 4" OE nt Inn is convergent ne Inn d) is…

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A: We can expand each function into its corresponding power series and then assess the limit by looking…

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A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Cauchy sequence not converge in R.A O .may not converge in set of rational numbers Q .B O converges…

A: One of the famous result in real analysis is that, Cauchy sequences converges in a complete metric…

Q: 2. Determine the convergence and divergence of the following Sequences: 3" 6" b. (1 -)") c. flan a.…

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Q: Consider these four statements: 1 A: 2 + 23+... +...+ 2.3 for every integer n > 1. n(n+1) n+1' B:…

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Q: The expansion 1 + x + x² ... approximates (1 - x)1 if O x is not equal to 1 本 O absolute value of x…

A: The expansion 1+x+x2+.........approximtes 1-x-1

Q: Cos 720 n! + A) diverges B) converges

A: if sum of series is finite then series is convergent. so we need to find sum of the series. note…