Consider the following statements:
- $b_{1}= \sqrt{2}$, series with each $b_{i}= \sqrt{b_{i-1}+ \sqrt{2}}, i \geq 2$, converges.
- $\sum ^{\infty} _{i=1} \frac{\cos (i)}{i^{2}}$ converges.
- $\sum ^{\infty} _{i=0} b_{i}$ converges if $\lim_{i \rightarrow \infty} \frac{|b_{i+1|}}{|b_{i}|} < 1$
Which of the following is TRUE?
- Statements $(1)$ and $(2)$ but not $(3)$.
- Statements $(2)$ and $(3)$ but not $(1)$.
- Statements $(1)$ and $(3)$ but not $(2)$.
- All the three statements.
- None of the three statements.