# is there a root test for the convergence of sequences (not for convergence of series)?

I've been having troubles finding information about the root test for sequences (not series!) that someone mentioned to me (I was interested in seeing a proof, at least). I've looked for it in many books and online, but could only find information about a root test for series. I have attached an image containing the test - is this genuine or is that person wrong?

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It's true. See these notes of Pete L. Clark for a proof. – David Mitra Feb 2 '14 at 16:52
By the way, note it's not giving a test for the convergence of $(a_n)$; it's just saying that if the limit of the ratios exists, then so does the limit of the roots and in this case the two limits are the same. – David Mitra Feb 2 '14 at 16:53
Thank you for your answer, David, but aren't the notes about series? I'm looking for information about the convergence of sequences (not series), I will edit my question to make that clearer. – 0ana Feb 2 '14 at 16:57
See Lemma 3 on page 3 (it applies to any sequence of positive terms). – David Mitra Feb 2 '14 at 16:58
I answered too soon, I should have read more carefully! Thank you so much, this answeres my question. Would you post it as an answer for me to be able to accept it? – 0ana Feb 2 '14 at 17:00

(By the way, note the result is not giving a test for the convergence of$(a_n)$; it's just saying that if the limit of the ratios exists, then so does the limit of the roots and in this case the two limits are the same. )