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Part I: Answer each of the following questions by encircling your choice [each 2 points].

`\sum_(n=1)^(\infty ) ((n^(2)+n)/(2n^(2)+1))^(n)`

converges.

`,T,F`

The series

`\sum_(n=1)^(\infty ) (n^(2)+1)/(n^(5)+n^(2)\sqrt(3))`

converges by Limit Comparison Test against the p-series

`\sum_(n=1)^(\infty ) (1)/(n^(3))`

. T

`,`

F The series

`\sum_(n=1)^(\infty ) ((-1)^(n))/(n)`

converges absolutely.

`,F`

The series

`\sum_(n=1)^(\infty ) ((n!)^(n))/(n^(2n))`

converges.

`,F`

The Maclaurin series of

`f(x)=ln(1+(1)/(2)x^(2))`

at

`a=0`

is given by

`\sum_(n=1)^(\infty ) (x^(2n+2))/(2^(n)(2n+2))`

.

`,T,F`

If

`\Sigma a_(n)`

is divergent, then

`\Sigma |a_(n)|`

is divergent.

`,T,F`

`\sum_(n=1)^(\infty ) arctan((n^(2))/(1+n))`

diverges.

`,F`

The radius of convergence of the series

`\sum_(n=0)^(\infty ) ((n+1)(x-2)^(2))/((2n+1)!)`

is

`R=\infty `

.

`,F`

Part I: Answer each of the following questions by encircling your choice [each 2 points].

`\sum_(n=1)^(\infty ) ((n^(2)+n)/(2n^(2)+1))^(n)`

converges.

`,F`

The series

`\sum_(n=1)^(\infty ) (n^(2)+1)/(n^(5)+n^(2)\sqrt(3))`

converges by Limit Comparison Test against the p-series

`\sum_(n=1)^(\infty ) (1)/(n^(3))`

. T

`F`

The series

`\sum_(n=1)^(\infty ) ((-1)^(n))/(n)`

converges absolutely.

`,F`

The series

`\sum_(n=1)^(\infty ) ((n!)^(n))/(n^(2n))`

converges.

`,F`

The Maclaurin series of

`f(x)=ln(1+(1)/(2)x^(2))`

at

`a=0`

is given by

`\sum_(n=1)^(\infty ) (x^(2n+2))/(2^(n)(2n+2))`

.

`,T,F`

If

`\Sigma a_(n)`

is divergent, then

`\Sigma |a_(n)|`

is divergent.

`,T,F`

`\sum_(n=1)^(\infty ) arctan((n^(2))/(1+n))`

diverges.

`,F`

The radius of convergence of the series

`\sum_(n=0)^(\infty ) ((n+1)(x-2)^(2))/((2n+1)!)`

is

`R=\infty `

.

`,F`

The Taylor Series for the following function

`f(x)=(1)/(2x)`

about

`a=0`

is

`(1)/(6)\sum_(n=0)^(\infty ) ((x+3)/(3))^(n)`

.

`,T,F`

If the interval of convergence of a power series

`\sum_(n=0)^(\infty ) c_(n)x^(n)`

is

`[-9,11)`

, then the radius of conver- gence of the series

`\sum_(n=0)^(\infty ) (c_(n))/(n+1)x^(n+1)`

is

`R=11.,T,F`