Determine Whether the Following Series is Convergent or Divergent By Applying the Relevant Tests: (a). 5+10+15+20+25+30+35+…+… | Engineering Mathematics 2 Assignment,, Singapore

Engineering Mathematics 2 Assignment Questions

Question 1.

(A). Determine whether the following series is convergent or divergent by applying the relevant tests:

(i). 5+10+15+20+25+30+35+…+…

(ii). ∑_(n=0)^∞=〖(-1)〗^n

(B). Evaluate the limit of the following sequence

(i). {(〖4n〗^2+5n+7)/(〖-8n〗^2+6n+5)}_(n=0)^(+∞)

(ii). {(n^2+7n)/(〖2n〗^3-12)}_(n=0)^(+∞)

(iii). {√((3n+2)/(12n-7))}_(n=0)^(+∞)

(C). Use the limit comparison test to determine whether the series , ∑_(n=0)^∞=1/√(n^2-9)is convergent or divergent.

Question 2.

(A). Find the centre, radius and interval of convergent for following power series: ∑_(n=0)^∞=1/(1+n^3 ) 〖(x+3)〗^n

(B). Find the radius and interval of convergent of power series ∑_(n=0)^∞=〖n(x+3)〗^n/4^(n+1)

(C). Find the Maclaurin series for e^x

(D). Find the first four terms of Taylor series for cos⁡x at x=3

Question 3

(A). Find ∂y/∂x and ∂z/∂x for x^4+y^4+z^4+x^2 y^2 z^2=0

(B). Find the Jacobian (∂(x,y))⁄(∂(u,v))

(i). x=2u+3v,y=u-3v

(ii). x=7u+v,y=2u+5v

(C). Suppose f(x,y,z)=x^3 y^2+xz+2z=3;

Given: x=2 sin⁡t;
y=2cos⁡t;
z=1;

(i). Find df/dt

(ii). Evaluate df/dt when t=0 for f (1,1,1)

Question 4

(A). Find the critical points of the following function, then determine whether they are the relative maximum, relative minimum or saddle points

f(x,y)=〖2x〗^3-〖27x〗^2+48x+〖2y〗^3-18y^2+48y+222

(B). Find the divergence and curl of the following vector fields:

(i). F(x,y,z)= xyzi+2y^2 zj+〖3z〗^2 k

(ii). F(x,y,z)= z sin y i+4xz j- cos 9z k