Strategy for Series — Question 5

*Question 5 Determine whether $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2+\cos n}$ converges. For the tail $n\ge2$, use only the bound $-1\le\cos n\le1$; no exact trigonometric values are needed.

*Question 5 – Solution

For $n=1$, $\cos1>0$ because $0<1<\pi/2$. For $n\ge2$, $n^2+\cos n\ge n^2-1>0$.

For $n\ge2$, $$n^2+\cos n\ge n^2-1\ge\frac12n^2.$$ Taking reciprocals reverses the denominator comparison: $$0<\frac1{n^2+\cos n}\le\frac2{n^2}.$$

The series $\sum 2/n^2$ converges because it is a constant multiple of a $p$-series with $p=2>1$.

The given series converges. A direct comparison is more efficient than ratio or root tests because the bounded trigonometric perturbation does not change the dominant $n^2$ denominator.