Question

nsin(pi/n), does this series converge or diverge , if it converges then what is the limit??

Answer 1

what have you tried so far ?

Answer 2

well, i input it into the calculator and it goes from 0 to 3.1 and that is it , it seems that doesnt go any higher than that. i can figure out how to solve it

Answer 3

Notice that sin(pi/n) dances back and forth between -1 and 1

Answer 4

so nsin(pi/n) ranges netween -n and n

Answer 5

as n-> infinity, clearly the value of nsin(pi/n) wont be 0.
what does that tell you about the series ?

Answer 6

ok so its an alternating series, but i checked the calculator and it does not alternate from negative to postive, they are all positive

Answer 7

its not alternating series

Answer 8

sin(pi/n) can take any value between -1 and 1
not just -1 and 1

Answer 9

find the limit and see what you get

Answer 10

\[\large \lim\limits_{n\to\infty} n\sin(\pi/n) = ?\]

Answer 11

it seems that there is no limit

Answer 12

\[\large \lim\limits_{n\to\infty} n\sin(\pi/n) = \pi\lim\limits_{n\to\infty} \dfrac{\sin(\pi/n)}{\pi/n} = \pi.1 = \pi \ne 0\]

Answer 13

so the series diverges

Answer 14

yes

Answer 15

the partial sums are positive and keep on increasing because sin(pi/n) is always positive for n > 1

Answer 16

i thought sin(pi/n) was inbetween 1 and -1

Answer 17

sin(x) ranges between -1 and 1

but lets look at sin(pi/n) for n>=1 :
n=1, sin(pi/n) = ?
n=2, sin(pi/n) = ?
n=3, sin(pi/n) = ?

Answer 18

evaluate them to see why sin(pi/n) is positive for n>1

Answer 19

1-0, 2-1 ,3-.866