Q3) Express cos(x²) as a power series. а) 1. 2! + 6! 4! x10 + 3! x14 +. 7! b) 1– 5! c) 1– x5 x7 + 7! + 3! 5! d) 1- 2! 4! 6! + +

Q: Which of the following represents any term of the sequence 6 12 3 5' 2 20 30 42 56 ... ? 1' 7 9 11'…

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Q: Determine whether the series is absolutely co Σ (-1)" + 1 n! n = 1

A: Given the series ∑n=1∞n65nn!(−1)n+1

Q: The sum of the series 1) is 1 n =1n+1 O A. -1 О в. 1 OC.O O D.1 O E. a 10 1/2 立

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Q: a) Find A and B such that (30)* (6*+1-5*+1)(6* –5*) 5* A 5*B 6*+1_5k+1 6*-5* b) Use the result in…

A: We can find the A and B values by partial fractions method

Q: Find the following infinite TM & α) Σ ntl (-3) n+1 A-1 5" infinite sums. 2 b) Σ (2n+1)(2n+3) n=1

A: Step 1: Step 2: Step 3: Step 4:

Q: 16. The power series +(x – 1) + 6(x – 1)² + 4(x – 1)³ + (x – 1)* + . is a Taylor series…

A: The given power series is x-1+6x-12+4x-13+x-14+... Compare with:…

Q: If z is a variable, then the infinite series of the form is called n Σ" =a + a₁ + ª₂x² + ... + ª₂x”…

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Q: Q.18) A series (-1)4+1 2k+1 , is: a) diverge by alternating series test c) converge by comparison…

A: I have appended the formula used in the solution part of the question

Q: The great Swiss mathematician Leonhard Euler sometimes reached incorrect conclusions in his…

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Q: Determine the value of the series E - n1 n² + 9n + 20 b)1 c)0 d) e)3

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Q: 7.- Prove that the series n=1 n³ n4 1 is divergent.

A: We have to prove the given series is divergent.

Q: 1.7 Show that 1² +2²+ +n² -1 n(n +1) (2n + 1).

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Q: (-1)" 2) The series n=1 is п a) Convergent, but not absolute b) Converges absolutely c) Diverges

A: Answer and explanation is given below...

Q: 3. Find the interval on which the curve of y= 2 1+3t+t² dt is concave up.

A: Second derivative is positive.

Q: Prove that x Auo +. (1– x)2 (1– x)3 4ot uj x+ uzx+ + : 1- x ... Hence find the sum of the series…

A: The sum of the terms in an infinite geometric progression is calculated using the formula: Sn=a1-r,…

Q: Prove that the series Σ Στ 11 n+ 7 4

A: Note: There is a typo in your question. The sum will be 7/4 if the series starts from n=0 to…

Q: Prove that the binomial series a(a – 1) a (a- 1)... (а — п) + | | (1+ x)ª 1+ ax + 2! n! +...…

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Q: Find the coefficient of x° in the power series of each of these functions. a) (1+x+x+x°+ .) b x +x…

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Q: Find the Maclaurin series for the given function. A) B) 6+z (-1)"z*+1 6 (-1)"z" o 6+1 D) (-1)"z"…

A: In this question we have to find the Maclaurin series of the function.

Q: 2 3 () is equal to 1+ (x – 1) 1) Show that the sum of the series 1+ 2 +.. +... (x – 1)2 + (x –- 1)³…

A: We have to solve given problem:

Q: 29. Which of the following is a power series expansion of 6 X 2n X x (A) 1 + + + + + 2! 6! (2n)! 6 X…

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Q: Express the sum 23 25 26 32 16 64 3n ()". (=) in terms of geometric series (hint: group Зп — 1 Зп —…

A: Given:

Q: Find a power series solution of y" + (1+x²)y=0 13 a. ao(1-x². - + x² + ...) + a₁(x-²1²x³- 24 720 13…

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Q: Determine the value of the series Σ 1 n2 + 7n + 12 T=1 a)1 d) e)0 1/4

A: Solution; Consider the given series; ∑n=1∞1n2 + 7n+ 12 or ∑n=1∞1n2 + 4n+3n+…

Q: 3) Find the laurent series expansion for: a) z² sin , 0 D 1-cosz (2-271)3 7 b) ======== on i) 1 2…

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Q: 1.5 Show that the series (a – 6)(a – 26) „3 + (a- b)(a- 26)(a- 35) | z+- 2! 3! 4! converges…

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Q: hat' H(x)= I ñ* ,x>1. (±) **9e グニン (a) Show that the series Convi unit on [a, as) where a> (b) Shaw…

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Q: um the following: (pa-py-kak k=0

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Q: n Expand x" in a series of Hermite polynomials.

A: We have to expand x^n in a series of  Hermite polynomials.

Q: a f(x) = √++ x b. fcx.)..z f(.x.)..=.! (1+X)4 f(x)= X 4+x² 2

A: The functions are given as a fx=1+xb fx=11+x4c fx=x4+x2 We need to expand these functions using…

Q: etermine the convergence or divergence of ∞ ΣΕ n=1 (−1)n+1 3η

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Q: Q2) The summation of ex series is x" a) En=0 n! b) Σ-01 5+n! xn+1 c) En=0 (п-1)! d)

A: Given:   Let fx=ex To find the Maclaurin series coefficients, we must evaluate          dndxnfxx=0…

Q: The first 4 terms of the Maclaurin series for In(1+3x) is ...? a) 3x - 2x² +9x³ - 81x4 Ob) 1+ 3x -…

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Q: ) Find the Taylor series centered at 1 for X - 2X? + 7

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Q: a) Find A and B such that (30)* (6*1 5*")(0* _5*) 5*A 5* B + b) Use the result in part a) to find a…

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