# Thread: Root test & Ratio test

1. ## Root test & Ratio test

2. Ratio test : first of all, if $\frac{a_{k+1}}{a_k} \rightarrow \infty$, then the sequence does not converge, which means the series $\sum a_k$ surely won't converge.

Same with the root test - if limsup $(a_k)^\frac{1}{k} \rightarrow \infty$, then $a_k \rightarrow \infty$, and the sequence diverges, so the series will diverge too.

Ratio test : first of all, if $\frac{a_{k+1}}{a_k} \rightarrow \infty$, then the sequence does not converge, which means the series $\sum a_k$ surely won't converge.
Same with the root test - if limsup $(a_k)^\frac{1}{k} \rightarrow \infty$, then $a_k \rightarrow \infty$, and the sequence diverges, so the series will diverge too.