Question

determine convergent or divergent

Answer 1

\[(\sin(n+1/2)\pi)/(1+\sqrt(n))\]

Answer 2

Is is a series?

Answer 3

yes

Answer 4

\[
a_n=\frac {\sin(n+1/2 \pi)}{1+\sqrt n}
\]
Is this the nth term?

Answer 5

yes, you got it

Answer 6

wait

Answer 7

the pi should be outside the parenthesis on the numerator

Answer 8

\[
a_n=\frac {\sin\left(\left(n+1/2\right) \pi\right)}{1+\sqrt n}
\]
Is this right?

Answer 9

well in my book pi is not apart of the sin funciton is multiplied by it i guess. like this sin(n+1/2)pi

Answer 10

i dont think I've ever seen it quite like that

Answer 11

\[
a_n=\frac {\sin\left(\left(2n+1 \right) \frac \pi 2\right)}{1+\sqrt n}

\]

Answer 12

In fact
\[
a_n =\frac {(-1)^{n+1}}{1+ \sqrt n}
\]
The series is an alternating series, and it is convergent by the alternating series test.