Ratio test : first of all, if , then the sequence does not converge, which means the series surely won't converge.
Same with the root test - if limsup , then , 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?