## anonymous 5 years ago ∑sin π/k converge or diverge? please attach you worksheet.

1. amistre64

doh!! and i was wrong lol http://www.wolframalpha.com/input/?i=%E2%88%91sin+%CF%80%2Fk

2. amistre64

i shoulda went with my first guess.... lol

3. watchmath

Nice problem! :D Notice that geometrically, the line passes through (0,0) and $$(\pi/2,1)$$ lies below the graph of $$\sin x$$ on the interval $$[0,\pi/2]$$. Hence on that interval we have $\frac{2}{\pi}x<\sin x$ It follows that for $$k\geq 2$$ we have $$2/\pi \cdot \pi/k<\sin (\pi/k)$$, i.e $$2/k<\sin(\pi/k)$$. But since $$\sum 2/k$$ diverges, then $$\sum \sin(\pi/k)$$ diverges as well.