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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?

Root test for sequences convergence?

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It's true. See these notes of Pete L. Clark for a proof. –  David Mitra Feb 2 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 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 at 16:57
    
See Lemma 3 on page 3 (it applies to any sequence of positive terms). –  David Mitra Feb 2 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 at 17:00

1 Answer 1

up vote 1 down vote accepted

It's true. See these notes of Pete L. Clark for a proof (Lemma 3 on page 3). Another proof can be found here at MSE.

(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. )

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Thank you, once again, for your fast and helpful answer! –  0ana Feb 2 at 17:48
    
@oana You're welcome. –  David Mitra Feb 2 at 17:50

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