### Determine whether the sequence converges or diverges.

Determine whether the sequence converges or diverges. If converges find the limit. A) an = (n + 2)!/n! B) {n^2e^-n} C) an = ln(2n^2 + 1)-ln(n^2 + 1) D) an = cos^2n/2^n

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Determine whether the sequence converges or diverges. If converges find the limit. A) an = (n + 2)!/n! B) {n^2e^-n} C) an = ln(2n^2 + 1)-ln(n^2 + 1) D) an = cos^2n/2^n

Please find attached problems to solve in ps2.doc file. The tables are in the tables.zip file. The solutions by another student are in the pdf file in the zip file attached. The other ms excel files are also there. Please check the solutions up. If the solutions are fine then you can submit a response with any comments or

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2. Arfken, p. 342, 5th Ed. (p. 359, 6th Ed.), ) Prob. 5.6.7. Use the General Vector Taylor Series Expansion For a General Function, cI) (r) = (I) (x, y, z) , Of a Three-Dimensional Vector Coordinate, Expressed In Cartesian Coordinates, which is Expanded About the Origin, r = 0 Or x = y = z = 0 , Where 0(0 = (1)(x', y', z') 1 ir,

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Determine the lower bound for the radius of convergence of the series solution about x y'' + 4y' + 6xy = 0, x=1

Determine whether or not each of the following limits exists. Discuss also the continuity of each of the following functions at given point c. Give reasons to your answers. Please see the attached file for the fully formatted problems.

Discuss the convergence and the uniform convergence of each of the following sequences of functions on the given set D. Give reasons to your answers: a) f_n(x) = (x/n)e^(-x/n), D = [0, infinity) b)f_n(x) = (x^n)/(1 + x^2n), D = [2, infinity)

Prove that if for each natural number, n, the function f_n on I = [0,1] ---> the reals is monotone increasing and if f(x)=limit as n--->infinity of f_n(x) is continuous on I, then the convergence is uniform on I.

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Jim Outfitters makes custom fancy shirts for cowboys. The shirts could be flawed in various ways, including flaws in the weave or color of the fabric, loose buttons or decorations, wrong dimensions, and uneven stitches. Jim randomly examined 10 shirts, with the following results: Shirt Defects 1 8 2 0 3 7 4 12 5 5 6 10

A) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity" ps. {An} is a sequence of non-zero complex numbers b) Prove that although the following power series have R=1 sum(nz^n) does not converge on any point of the unit circle.

(See attached file for full problem description) evaluate each limit Evaluate:

Prove the following statement using the epsilon-delta definition of limit. lim lxl = 0 x-->0 Thank you so much for you help!!!

Given f(x) = x^2 + 3, find the exact area A of the region under f(x) on the interval [1, 3] by first computing n Σf(xi)Δx and then taking the limit as n-->∞. i=1 Please see the attached file for the fully formatted problems.

1. For the equation ?(x)= x^(1/2) a) Find the Taylor polynomial of degree 4 of at c = 4 b) Determine the accuracy of the polynomial at x = 2. 2. Find the Maclaurin series in closed form of a) ?(x)=((1) / ((x+1)^2) b) ?(x)=ln ((x^2)+1) 3. Use the chain rule to find dw / dt, where w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(

1.) one of the following series converges absolutely one converges conditionally, the other diverges which is which show steps a.)sum (-1)^n (1/(sqrt(n))) b.) sum (-1)^n (sqrt(n)/(1+sgrt(n))) c.) sum (-1)^n (1/( n(1+sqrt(n))) 2.) test for convergence the sum of ((2n+n^3)/(3+n^4))

1.) list the first 4 terms of the series from n=1 to infinity for (1*3*5***(2n-1))/ (n^2 * n!) 2.) determine the interval of convergence of the power series sum of ((2^n)/(1+n)) * x^n

By manipulating the geometric series for the sum of n=0 to infinity for x^n=1/(1-x) for |x| < 1 determine the power series (about 0) for x/(2+3x)

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1.) use integral test p>1 for series sum of n/((1+n^2)^p) to show that it converges using substitution 2.) a.) explain why sum from n=0 to infinity of (-1)^n (n^2/(1+n^3)) converges b.) how many terms of that series should you sum to have an error no more than (1/100)

(See attached file for full problem description) Evaluate the following limits (explain how you solved the limits)

1.) find maclaurin series for f(x) do not show rn(x)->0 also find radius convergence of f(x)= ln(1+x) 2.) find the taylor series for f(x) centered at the given value of a .. assume a power series expansion, do not show rn(x)-> 0 of f(x)=sin x , a=pie/2 3.) find the sum of the series sum from n=0 to infinity of (3^n

Prove theorem 7.3 in notes attached.

Find the radius of convergence and interval of convergence of the series of the following two problem 1.) sum n=1 to infinity of (n^2*x^n)/(2.4.6.......(2n)) 2.) sum n=1 to infinity of ((2.4.6........(2n))/(1.3.5......(2n-1)))x^n keywords: radii, intervals

Show that a countable set in R^n is of measure zero. Notes for this section are attached. keywords: measurable, measurability

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If f is a function from R to R, and there exists a real number aE(0,1) such that |f'(x)|≤a for all xER , show that the equation x = f(x) has a solution.

Assume that g(t) is continuous on [a,b], K(t,s) is continuous on the rectangle a≤t, s≤b and there exists a constant M such that (a≤s≤b). Then the integral equation has a unique solution when . Please see the attached file for the fully formatted problems.