<s>[INST] Finding a convergent majorant series 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:
$$
|f_n(t)| = |\frac{1}{n^3} \cos(nt) - \frac{1}{(2n+1)^2} \sin(nt)| \leq \frac{1}{n^3}
$$
 [/INST] 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 $|\sin (nt)|\leq 1$.
</s>