# (Solved): Part I: Answer each of the following questions by encircling your choice [each 2 points]. \sum_(n=1 ...

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)

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

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