has a convergent subsequ lim an = x, then n4x 4. The sequence {an}, where {an = n+1 5. Let x R. If there exists a sequence {an} CACR such that of A. 6. Let D={100: z € Z}. Then D is a dense subset of R.

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Q: Problem 5: Prove Vn EN that 6(7") — 2(3") is divisible by 4. -

A: Let P(n) = 6(7n) - 2(3n) is divisible by 4  Step 1 : let n = 1  P(1) = 6(71) - 2(31)  = 42 - 6 = 36…

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Q: 8. cos mt sin nt (m ‡n)

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Q: Evaluate by the residue theorem: J° sind. Use the formula 4 sin³ r = 3 sin x - sin 3x].

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Q: Evaluate by the residue theorem: foodz, 0<x< 1 1+x sin(an)

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Q: Solve the initial value problem with y(0) = 2. y = dy dt y= = 1 exp(t) + 21 exp(8t)

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Q: EVALUATE: A = [² [²√² + ¹ dydr 1 X HINT: Perform a reversal of integration

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Q: 7. Let x ER and A be a nonempty subset of R. If x is not a limit point of A then x is an isolated…

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Q: Determine L ¹{F}. F(s) = -s²-2s+5 (s+ 2)² (s+3)

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Q: ax + by + c > 0,

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Q: = ant(n-1) ·+an-kt(n.

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Q: Consider the function f(x) = cos(2x). 1. Find the Taylor series for f(x) at x = n.

A: Due to Bartleby answering guidelines I have solved only first question which is question no 1.

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Q: (b) Consider the set Prove that inf A = 0. A = { n m² : n,m € Z¹, m≤n}.

A: Our objective is given below:

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A: I have explained everything with in the solution .please go through that.

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