`cos(pi n)={(1 if n=2k-1),(-1 if n=2k):},` `k in ZZ` i.e. it is equal to `1` for odd `n` and `-1` for even `n.` Therefore, we can break this into two cases.

`n=2k-1` **(n is odd)**

`lim_(n to infty)a_n=lim_(n to infty)1/n^2=1/infty^2=1/infty=0`

`n=2k` (n is even)

`lim_(n to infty)a_n=lim_(n to infty)-1/n^2=-1/infty^2=-1/infty=0`

**Since the limit is the same in both cases, the sequence is convergent and its limit is equal to zero.**

Image below shows first 15 terms of the sequence. We can see that the odd-numbered terms are negative, while the even-numbered terms are positive, but they are both approaching the `x`-axis implying convergence to zero.

**Further Reading**