You need to ask if converges.
I need to find radius of convergence and interval of convergence of
In d'Alembert feature I find radius like this
Radius is 1
If I try in Cauchy feature, I get same:
first square root
square root
Radius is 1
Interval of convergence:
I take -1 and 1
and
I do check if diverge/converge:
it diverge
here != means 'not even'
Interval is
Questions:
- Did I do task correct?
- it diverge or converge?
- How interval of convergence change because of divergence and convergence?
It doesn't have limit, but it still get two answers: 1 and -1. If it's like this, it is still said that it converges?
How does answer change because of that?
How would answer change if it converged with limit 1/2?
And how would answer change if it would converge with limit 2?
By the ratio test, the series converges where the limit , diverges where the limit is and inconclsive where the limit .
So all we can say at the moment is that the series is convergent when , because this is where the limit .
You will need to use a different test to test the endpoints.
Well of course it is. You're testing if the function will converge at the endpoints. So obviously you need to see if the endpoints converge or diverge.
Specifically for the case of when the series becomes , this is a p-series with . What do you know about p-series?
For the case of when the series becomes , what are the conditions for which Leibnitz's Alternating Series Theorem implies convergence?