- #1

- 1,270

- 0

## Main Question or Discussion Point

Let ∑a

Suppose a

If c<1, then ∑a

If c>1, then ∑a

If c=1, the test is inconclusive.

What if a

Suppose limsup (a

If r<1, then ∑a

If r>1, then ∑a

If r=1, the test is inconclusive.

What if limsup (a

Thanks for clarifying!

_{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!