Originally Posted by
Drexel28 It should be $\displaystyle \frac{1}{x}+\frac{1}{3x^3}+\cdots$. It's the Maclaurin series for $\displaystyle \tan(x)$.
Alternatively, suppose that $\displaystyle \frac{1}{n}-\tan\left(\frac{1}{n}\right)$ shared convergence with some series $\displaystyle \frac{1}{n^\lambda}$. Then, we get that $\displaystyle \lim_{\infty}\frac{\frac{1}{n}-\tan\left(\frac{1}{n}\right)}{n^\lambda}=C$ for some constant $\displaystyle C$. Letting $\displaystyle z=\frac{1}{n}$ gives $\displaystyle \lim_{z\to0}\frac{z-\tan(z)}{{\color{red} z^\lambda}}=C$. Try to find a $\displaystyle \lambda$ that works.