# Math Help - 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.

3. Originally Posted by adam63
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.
So it looks like the root test and ratio test is applicable for the case c=∞ and r=∞ as well. (and we can say c=∞>1 or r=∞>1, hence the series diverges)

But I don't understand why your claims are true. Can you explain/justify?

Thank you!

4. a series can converge ONLY if the sequence converges.
Then, if the sequence diverges, the series will surely diverge .