(Solved): (d) Use Leibnitz's rule of differentiation under the sign of integration to show that \int_0^(x^(2)) ...

(d) Use Leibnitz's rule of differentiation under the sign of integration to show that

\int_0^(x^(2)) tan^(-1)((y)/(x^(2)))dy=(\pi -2log2)(x^(2))/(4)

(b) Find the surface area of the cylinder

x^(2)+y^(2)=a^(2)

cut out by the cylinder

x^(2)+z^(2)=a^(2)

.

5+5

(a) Define convergent sequence. Prove that a sequence in

R

can have at most one limit. (b) Discuss the convergence of the geometric series

1+r+r^(2)+cdots

for different volues of

r

.

2+4+4

b. (a) State and prove Cauchy's root test of the series

\sum_(n=1)^(\infty ) u_(n)

. (b) Test the convergence of the series

(1)/(2)+(2)/(3)x+((3)/(4))^(2)x^(2)+((4)/(5))^(3)x^(3)+cdots\infty ,(n

)

>

(

0) 2+4+4

(a) State Cauchy's first theorem on limits of the sequence

{x_(n)}

and hence show that

\lim_(n->\infty )[(1)/(\sqrt(n^(2)+1))+(1)/(\sqrt(n^(2)+2))+cdots+(1)/(\sqrt(n^(2)+n))]=1

(b) Determine the radius of convergence and the exact interval of convergence of the power series

\sum (1*2*3cdotsn)/(1*3*5cdots(2n-1))x^(2n)

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