Showing divergence for $\sum^\infty_{n=1} \bigl(1-\cos (1/\sqrt n)\bigr)$

Question

Show the divergence of $\displaystyle\sum^\infty_{n=1} \left(1-\cos\frac 1 {\sqrt n}\right)$

My attempt:

Since $\sin x\in [-1,1]$ then $\sin\frac 1 {\sqrt n}-\cos\frac 1 {\sqrt n}\le 1-\cos\frac 1 {\sqrt n}$

It's easy to see that $\sin\frac 1 {\sqrt n}-\cos\frac 1 {\sqrt n}$ doesn't have a limit so its series must diverge (is that correct in general?) thus from the comparison test the given series diverge as well.

Note: no integrals, nor Taylor, nor Zeta.

Answer 1

Hint: $$\left(1-\cos\frac 1 {\sqrt n}\right)=2\sin^2 {1\over 2\sqrt n}$$ Now use the limit comparison test with the harmonic series $\sum {1\over n}$

Answer 2

But series converge $$\sum _{j=1}^{\infty } \left(1-\cos \left(\frac{1}{j}\right)\right)=0.778759$$