Home /
Expert Answers /
Advanced Math /
d-use-leibnitz-39-s-rule-of-differentiation-under-the-sign-of-integration-to-show-that-int-0-x-2-pa707

(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)`