Test the convergence of the series: sum (1 to inf) tan(1/k^2).

Test the convergence of the series:

sum(1 to inf) tan(1/k^2).

My teacher gave a hint that says "recall from calc 1 how to calculate the lim (x->0) tanx/x without using L'Hospital's Rule. If I use L'H then I will not get credit for the problem so I have to take his "suggested" hint and use it. So I have no idea what to do. I was going to use the ratio test and compare to 1/k^2. Now with his hint I am not sure what to do. Thank you for your help!!!

Use the limit comparison test by comparing with sum (1 to inf) 1/k^2.

lim k-->infinity tan(1/k^2) / (1/k^2)
= lim x-->0 tan(x)/x using x=1/k^2
= lim x-->0 (sin(x)/cos(x)) / x
= lim x-->0 (sin(x)/x) / cos(x)
= [lim x-->0 sin(x)/x] / [lim x-->0 cos(x)]
= 1/1, since lim x-->0 sin(x)/x = 1, cos(0) = 1, and cosine is continuous
= 1, which is finite and nonzero.

So sum (1 to inf) tan(1/k^2 and sum (1 to inf) 1/k^2 both converge or both diverge.
sum (1 to inf) 1/k^2 converges from the p-test since 2>1.

We conclude that sum (1 to inf) tan(1/k^2) converges.

Lord bless you today!