1. Use any method for determining the convergence/divergence of the following (but the method must have rigor). Our methods include inequality comparison, limit comparison and integration itself. (d) E (In(n))² n2

Q: 1. Let z, be the sequence of approximations to the root of f(x) = 0 and r, is a suitably close…

A:

Q: 2. Gauss Iterative Method 4x + y + 3z = 17 x + 5y + z = 14 2x – y + 8z = 12 Assume a convergence of…

A: The objective  is solve the system of equations by Gauss iterative method

Q: 8. Consider the three functions a(x), b(x), and c(r) given by: a(x) =E b(x) = g3n+1 Σ g3n+2 Σ c(x) =…

A:

Q: Given the function f (x) = x³ + 4x² – 10, suppose that a = 1.5 is near to a root of f (x). i. Show…

A: Fixed point iteration method: To find the approximate solutions of the equation f(x) = 0,  rewrite…

Q: 4. Consider the sequence of functions : f₁(x) = 1 n³ [x−²+ +1 (a) the uniform topology (If yes, to…

A: The given sequence of function is ;    fn(x) = 1n3x-1n2+1 We have to answer the following questions…

Q: Find the radius of convergence for En-o(-1)" (2n + 1)z".

A:

Q: If a numerical method has global truncation error of order h³ , then reducing h by half, we expect…

A: Given that a numerical method has global truncation error of order h3, then by reducing we expect…

Q: 6. Find the radius of convergence and the interval of convergence for E-0¹ (x-2)".

A: 6. The given power series is     ∑n=0∞12nx-2n We need to find the radius of convergence and the…

Q: Show that the sequence f.(x) = xn converges for each xE[0,1] but that the convergence is not…

A:

Q: Determine the radius and interval of convergence of F (-1)*(x – 4)k k2k k=1

A:

Q: Given that lim n2(x- 5)" converges by the Ratio test. If x€ Z then the numerical n- 00 value of x is…

A:

Q: Find the radius of convergence and the interval of convergence. (x – 3)* 2k k=0

A: The given series is  ∑k=0∞ (x-3)k2k

Q: If E C, (x – 5)" converges at x = 11 and diverges at x = 15, what can you say about: (a) the…

A: Find the value of R and proceed. Find the center as clearly here it is 5 and proceed just as shown…

Q: In the graph below, 0<f(x)<g(x) with f(x) in red and g(x) is blue. What, if any, information do we…

A: This is a problem of Improper Integral.

Q: Assume that |xk+1 x* lim k→∞ xk - x*|r = C > 0, here x* is such that f(x*) = 0. This means that the…

A:

Q: (-1)"-1 Vn n=1 Input C for convergence and D for divergence:

A:

Q: 5. Use the comparison test to determine if each integral converges or diverges b) 2 2/3-4 a) e-x² dx

A: Solution: 5.a ∫1∞e-x2dx

Q: Question: 4(a). Use root test to determine convergence/divergence of the given E(-1)k+1(k +…

A:

Q: Find the interval of convergence of E(-1)". n 2n 36" n=0

A:

Q: Discuss the Convergence or Divergence for D { [4+ (-0"]" (Z+2)" こう 2n+1 - DZ² 2 لع Wi ngo (2n+¹)!

A:

Q: Determine the interval and radius of convergence for E-1) (+1)*

A:

Q: Suppose we have the sequence of functions $f_n(x)=x^n$ defined on $[0,1],$ and suppose $f_n\to f$…

A:

Q: х Zn=0 п! x2n Ση-0

A:

Q: Find the interval of convergence of E 27" (x – 7)3n+2 n=0

A:

Q: Find the radius and interval of convergence of the 00 (x- 2)" Σ (2n + 1)! n= 0

A:

Q: 1) Determine the interval and radius of convergence of (x-2)" (n+1)5*** 8. Σ n=0

A:

Q: Q2: If the positive root of the equation f(x) = ax + x- 1 is calculated using the fixed point method…

A: Let's find.

Q: 1 3. Suppose f(x) = En-o x" .Assuming convergence, what is n=0 n! f'(x)?

A:

Q: Find the interval of convergence of E n(x – 8)" n=2

A:

Q: 1. Find the interval and radius of convergence 00 (4x – 2)" (-1)" (4n + 1) n=0

A:

Q: Find the radius of convergence for: (n!)"" 2 TL (2n)! 1.

A:

Q: By using the substitution x = t² determine the convergence of d.x [ (x + 1)√x

A:

Q: 1. Find the radius of convergence of (-x)³n 8n (2n + 1)* n=0 In addition, analyze its convergence at…

A: Given series is ∑n=0∞-x3n8n(2n+1)

Q: 1. Show that fn(x) →0 for each x E [0,1]. -> 2. Determine whether or not the convergence is uniform.

A: In this question, we find the limit of the sequence of function and in the second part, we use Mn-…

Q: Find the interval of convergence ofS f(x) dx where f (x) = (-1)+(x-6)" %3D E

A: Given problem is :

Q: determine the radius and convergence of E n=1 5xn/3n2

A:

Q: 00 (-1)*+1 7n2 + 2 n=1 Input C for convergence and D for divergence:

A: Here,

Q: 1 5. Determine by the limit comparison test if Zn=2 converges or diverges. Vn-1

A:

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: Q2: If the positive root of the equation f(x) = ax +x-1 is calculated using the fixed point method…

A:

Q: The interval of convergence of E=1 (x+3)" is n2 -4 < x < -2 the above None of the other options

A:

Q: 7. Let x₁ = a > 0 and Xn+1 = x + 1/x,, for n E N. Determine whether (x,) converges or diverges.

A: Q7 asked and answered.

Q: Find the radius of convergence for п! Σ x". 1· 3· 5. ... · (2n – 1) n=1

A:

Q: O Discuss the convergence of f,(x)=x + , on A =[0,00).

A:

Q: The function f(x)= x' - 5xr² + 7x - 3 has a root at x = 3, write one re-arrangement to that root,…

A: For any point x0, consider the recursive process xn+1=gxn where n=0,1,2… If f is a continuous…

Q: 1 i. Use the result in (i) to test for convergence 2n =1 2 Vn? +5

A: See the attachment

Q: b. Si x-2ª 3 1

A: ∫13dxx-2    =  [ lnx-2 ]13   =  ln|3 - 2| - ln|1 - 2| =  ln |1| - ln |-1|   =  0 - 0  = 0

Q: 7. Let f(x): Find the interval of convergence of f. (2n+1)!

A: Given series