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Real Analysis

Convergence or Divergence of a Sequence

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges find its limit. an = (n-2)!/n! See the attached file.

Convergence of Power Series

1 Determine whether the series converges absolutely, converges conditionally, or diverges. ∞ Σ 2∙4∙6∙∙(2n)/2ⁿ(n+2)! n=1 2 Calculate sin 87° accurate to five decimal places using Taylor's formula for an appropriate functio

Power Series Proof

Define the set R[[X]] of formal power series in the indeterminate X with coefficients from R to be all formal infinite sums sum(a_nX^n)=a_0 +a_1X+a_2X^2+... Define addition and multiplication of power series in the same way as for power series with real or complex coeficients,i.e extend polynomial addition and multiplication t

Limits and Indeterminate Forms

Evaluate the limit, using L'Hospital Rule if necessary. lim sinax/sinbx as x approaches 0 lim ln x^4/x^3 as x approaches infinity

Construction of the air-conditioning system with heat load

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

Vectors and Limits

See attached file for full problem description. 1) Find the canonic equation of the straight (recta) in the space( espacio), with R 2) Calculate the scaling product of the vectors r y s knowing that r = 3i + 4j; that the module of the vector / s/ = 6; and the angle that both vectors form is 120° 3)Find

General Vector Taylor Series Expansion: Measure of deviation

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,

Taylor Series Expansion and derivation of the Euler Formula

Show that (a) sin x = summation (0-infinity) (-1)^n x^(2n+1)/(2n+1)! (b) cos x = summation (0-infinity) (-1)^n x^(2n)/(2n)! Use the Taylor series expansion around the origin, f(x) = summation (0-infinity)[x^n/n!]f^n(0), and derive the power series expansions for sin x , cos x and e^x. Then write out the first few real

Semi-Continuous Function on a Compact Subset

Show that if f is an upper semi-continuous function on a compact subset K of R^p with values in R, then f is bounded above and attains its supremum on K. Edit: R^p--Let p be a natural number and let R^p denote the collection of all ordered "p-tuples"---i.e. (x_1, x_2,..., x_p) with x_i being a real number

Stone-Weierstrass Approximation Theorem

Show that there is a countable set F of functions of form x--->a_0 + a_1cosx + a_2cos(2x) + ... +a_ncos(nx) such that every continuous real-valued function on [0,pi] is the uniform limit of a sequence of functions (f_n) in F.

3-Sigma Control Limits

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

Limits : Proving the Root Test Assuming the Ratio Test

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.

Evaluate limit

Evaluate each limit by the substitution approach. If 0/0 is produced, simplify the expression by factoring and then try again. (see attached) lim x^2 - 3x + 2 x->1 _____________ x - 1

Evaluating Limits

Evaluate each limit by the substitution approach. If 0/0 is produced, simplify the expression by factoring and then try again. lim (x+1)^1/2 x-->3

Proof: Sequences and Limits Example Problem

If {sn}∞ n=1 is a sequence of real numbers such that sn ≤ M for all n and lim n--> ∞ sn =L; prove that L ≤ M. Is the statement true if we replace both inequalities with "<"? See attachment for full equation.

Finding Area Using Sums and Limits

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 &#931;f(xi)&#916;x and then taking the limit as n-->&#8734;. i=1 Please see the attached file for the fully formatted problems.

Problem set 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. Question (2) Find the Maclaurin series in closed form of Question (3) Use the chain rule to find dw / dt, where w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(e^t) sin t, z=(e^t), t=0 Question (4): Find the critical points and test for relative extrema: ?(x,y)=2(x^2)+2xy+(y^2)+2x-3

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=(

Convergence and divergence determination

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))

Series for Infinity Terms

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