thank you this is what I have donePeroK said:
The radius of convergence is an interval centred on ##0##. It cannot be ##[-1,7]##.Amaelle said:
the exercice ask about the set of convergence and not the radius and as you can see we have y=x+3 y is centred in 0 not xPeroK said:
That's not what you have. You have ##y## centred on ##3##:Amaelle said:
Amaelle said:
You asked for a hint and the hint was to think about the radius of convergence.Amaelle said:
The width of the interval of convergence is two times the radius of convergence.Amaelle said:
Neither x nor y "converges in" some point or some interval. It's the series that converges, not values of x or y.Amaelle said:
That phrasing bothers me as well. I agree that it should be "converges AT x =4" or wherever.FactChecker said:
This sounds wrong. Are you thinking that "converges in x=4" means that the radius of convergence is 4? That is not how I would interpret it. I would guess that the phrase was bad and they meant "converges at x=4", which makes it converge for (x+3)=7. So the radius of convergence is 7. That will change your endpoints of the region of convergence.Amaelle said:
The convergence of a series refers to the property of a series where the sum of its terms approaches a finite value as the number of terms increases.
The convergence of a series can be determined by evaluating the limit of the partial sums of the series. If the limit exists and is a finite value, then the series is said to be convergent. If the limit does not exist or is infinite, then the series is said to be divergent.
Absolute convergence refers to the convergence of a series where the absolute value of each term is considered, while conditional convergence refers to the convergence of a series where the sign of each term is taken into account. A series can be absolutely convergent but not conditionally convergent, or it can be both absolutely and conditionally convergent.
Some common tests for determining the convergence of a series include the comparison test, ratio test, root test, and integral test. These tests involve comparing the given series to a known convergent or divergent series or evaluating the limit of the series using various techniques.
No, a divergent series cannot have a finite sum. The definition of convergence of a series requires the sum of the terms to approach a finite value, so a series that does not meet this criteria is not considered convergent.