Finding a convergent majorant series

Question

I have a series $$ \sum_{n=1}^\infty \left( \frac{1}{n^3} \cos(nt) - \frac{1}{(2n+1)^2} \sin(nt) \right) $$ and I have to find a majorant series to this series.

The convergent majorant series I was supposed to find is $$ \sum_{n=1}^\infty \left( \frac{1}{n^3} + \frac{1}{(2n+1)^2} \right). $$ however, I don't exactly get why.

I thought that $\frac{1}{n^3}$ is also a convergent majorant series.

I know that the first term $\frac{1}{n^3} \cos(nt)$ is always less than or equal to $\frac{1}{n^3}$ and subtracting the second term $\frac{1}{(2n+1)^2} \sin(nt)$ will only make it smaller, so why doesn't this inequality hold: $$ |f_n(t)| = |\frac{1}{n^3} \cos(nt) - \frac{1}{(2n+1)^2} \sin(nt)| \leq \frac{1}{n^3} $$

Answer 1

Hint. You may use $$ |a+b|\leq |a|+|b|,\qquad a,b \in \mathbb{R}, $$ giving, for $n=1,2,3 \cdots,$ $$ \left| \frac{1}{n^3} \cos(nt) - \frac{1}{(2n+1)^2} \sin(nt) \right|\leq \left| \frac{1}{n^3} \cos(nt)\right|+\left| - \frac{1}{(2n+1)^2} \sin(nt)\right|\leq \frac{1}{n^3} + \frac{1}{(2n+1)^2} $$ since $|\cos (nt)|\leq 1$ and $|\sin (nt)|\leq 1$.