Question

does this series converge or diverge?
sum from n=1 to infinity of log(n*sin(1/n) )

Answer 1

@hartnn @dan815

Answer 2

$$
\Large{\sum_{n=1}^{\infty} \log(n\cdot \sin(1/n) )
}$$

Answer 3

By the comparison test, the series converges.

Answer 4

do you agree that
sin(1/n) < 1/n

Answer 5

it is true that sin x < x for x > 0 , where x is in radians

Answer 6

remember, arclength = radius * theta .
Here radius is 1 . theta I labeled as x

Answer 7

you can see that geometrically, the arc of length x must be longer than sin x .