- #1
kingwinner
- 1,270
- 0
Let ∑ak be a series with positive terms.
Ratio test:
Suppose ak+1/ak -> c.
If c<1, then ∑ak converges.
If c>1, then ∑ak diverges.
If c=1, the test is inconclusive.
What if ak+1/ak diverges (i.e. ak+1/ak->∞)? Do we count this as falling into the case c>1? Can we say whether ∑ak converges or not?
Root test:
Suppose limsup (ak)1/k = r.
If r<1, then ∑ak converges.
If r>1, then ∑ak diverges.
If r=1, the test is inconclusive.
What if limsup (ak)1/k = ∞? Do we count this as falling into the case r>1? Can we say whether ∑ak converges or not?
Thanks for clarifying!
Ratio test:
Suppose ak+1/ak -> c.
If c<1, then ∑ak converges.
If c>1, then ∑ak diverges.
If c=1, the test is inconclusive.
What if ak+1/ak diverges (i.e. ak+1/ak->∞)? Do we count this as falling into the case c>1? Can we say whether ∑ak converges or not?
Root test:
Suppose limsup (ak)1/k = r.
If r<1, then ∑ak converges.
If r>1, then ∑ak diverges.
If r=1, the test is inconclusive.
What if limsup (ak)1/k = ∞? Do we count this as falling into the case r>1? Can we say whether ∑ak converges or not?
Thanks for clarifying!