Let ∑a(adsbygoogle = window.adsbygoogle || []).push({}); _{k}be a series with positive terms.

Ratio test:

Suppose a_{k+1}/a_{k}-> c.

If c<1, then ∑a_{k}converges.

If c>1, then ∑a_{k}diverges.

If c=1, the test is inconclusive.

What if a_{k+1}/a_{k}diverges (i.e. a_{k+1}/a_{k}->∞)? Do we count this as falling into the case c>1? Can we say whether ∑a_{k}converges or not?

Root test:

Suppose limsup (a_{k})^{1/k}= r.

If r<1, then ∑a_{k}converges.

If r>1, then ∑a_{k}diverges.

If r=1, the test is inconclusive.

What if limsup (a_{k})^{1/k}= ∞? Do we count this as falling into the case r>1? Can we say whether ∑a_{k}converges or not?

Thanks for clarifying!

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# Root test & Ratio test

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