Question

Why is ∑ n=2 ∞ cosnπ N − − √ alternating series whilst ∑ n=1 ∞ SinN N 3 Click to see better view of problem

Answer 1

Why is \[\sum_{n=2}^{\infty}\frac{ \cos n \pi }{ \sqrt{N} }\] alternating series whilst \[\sum_{n=1}^{\infty}\frac{ Sin N }{ N^{3}}\] is not

Answer 2

\[\cos 2\pi=1, \cos 3\pi=-1, \cos 4\pi = 1, \cos 5\pi = -1,...\] do you see now why the first series is an alternating series?

Answer 3

\[\sin 1 = 0.841, \sin 2 = 0.909, \sin 3 = 0.141, \sin 4 = -0.757\] clearly the terms of the series do not alternate in signs

Answer 4

but isn't the graph of sine and cos both have alternating values?