"Calculate the exact value of the sum using the binomial theorems 8 = Σ ([8], [n])( - 1)” 3″ - 8 3¹ n = 0 8 8 *= (*) (-1)*3*-* S= Σ n n=0

Q: 4. Find the power series representation of the form Σ=o arn where f(x) 7 2 + x f(x) = a) Σ n=0 7·χη…

A:

Q: 2 4 8 n2 + 2n 1 (i) 1 3 (ii) (iii) ) - 9 27 1+ 2n2 ' n=0 2nn n=1

A: Both series we can evaluate using geometric series method.

Q: 11. Find the Maclaurin series for | cos (2x³) dx. OS

A: here we have to find the intregration

Q: (d) Given that n is even, how many 00-free bitstring of length n where n is even have exactly (끌 +…

A:

Q: 8. Give a closed form for the sum You may assume that Σk=1 k = n(n+1) 2 1x2+2x3+ ... k(k + 1) and…

A:

Q: Find Maclaurin series for the following functions

A: “Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Using the binomial expansion n (1 + x)² = ± (1) x², k=0 prove the following relations n n…

A:

Q: 00 Σ (n + 1)²n (1 – n + 2n²)" n=1

A: We have to check whether given series is convergent or divergent.

Q: 1.13. Expressing the binomial coefficients in terms of fac- torials and simplifying algebraically,…

A: From the above given data the following solution is provided below:

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: Any number in radix-r with n digits in the integer part and m digits in the fractional part can be…

A: (a) To expand the summation in given equation for n=5, and m=4 ∑i=0n−1airi⋅∑i=1mair−1

Q: Use the binomial theorem to expand the following: (a) (x + 2y)* (c) (3x + 1)4 (b) (x - y)6 (d) (x +…

A:

Q: f(x) = (2+ kæ), |ka| < 2, where k is a positive constant %3D The binomial expansion of f(x), in…

A:

Q: x[n]=(0,7)n.u[n] and h[n]=(0,9)n.u[n] to be y[n]=x[n]*h[n] Get the result of my processing…

A: Solution is in step 2

Q: 13. (a) Express each of following sum in sigma notation 1 a) 1 + 2 + 113 +... + 1 20 b)…

A: As per our guidelines we can answer only 1 question so kindly repost the remaining questions…

Q: Evaluate 4 (1). 4

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: f(x+h)-f(x) 5) Simplify the difference quotient for f(x) = x+ using the Binomial Theorem. h Show…

A: Quotient using binomial expansion

Q: 3n+2 4n–2° n=1

A:

Q: For an n × n matrix A, we can similarly define the matrix exponential aseA = I + A + (1/2!)A^2 +…

A:

Q: 5. Let Ch k! = (k-n)!n! denote the binomial coefficient. (a) Prove that Ch k(n) = n!

A:

Q: 4. Prove that for all n E N η Σ( r=0 = = (3) Hint: Examine the coefficient of x² by expanding both…

A:

Q: If C'(x) ME = n=0 (2n)! 2n x²n and Xx ∞ S(x) = Σ n=0 (2n + 1)² series of C(x) + S(x). 2n n=0 (2n) 2…

A:

Q: 9. (a) Show that if n is an integer, then: -o3* (") = 4". %3D (b) Let n be a positive integer. Show…

A:

Q: Are there any other ways of decomposing 1/11 into sum of two unit fractions? In how many ways can we…

A: Decomposing Unit FractionsThe problem of decomposing a unit fraction as the sum of two other unit…

Q: = (1/2) + (1/6) + (1/12) + .... Write this in sigma summation notation

A: Given query is to write the expression in Sigma notation.

Q: 9. The Maclaurin series for the function * f(x) = sin²(x) 00 (-2)" (2x)2n+1 (2n + 1)! n=0 (-1)2a+1…

A: