find the radius and interval of convergence for the series
the series from n=1 to infinity of
((-1)^(n+1)*x^n)/n!
I did the ratio test so I had the Lim as n approaches infinity of -x/(n+1), but this is 0, giving no radius, so I think I did something wrong...
To find the radius and interval of convergence for the series:
We'll start by applying the ratio test correctly. The ratio test states that for a series ∑(an), if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. Otherwise, if the limit is greater than 1 or infinite, the series diverges.
Applying the ratio test to your series:
lim (n→∞) |((-1)^(n+1)*x^(n+1))/((n + 1)!)| / |((-1)^(n+1)*x^n)/n!|
Simplifying the expression, we get:
lim (n→∞) |x| / (n+1)
As you correctly found, taking the limit as n approaches infinity gives us 0. This means that the series converges for all x.
However, we haven't determined the radius of convergence yet. The radius of convergence (R) is the distance from the center of the interval (usually 0) to the point where the series converges. In this case, as the series converges for all x, the radius of convergence is infinite and we can say that the series converges for all real numbers.
In conclusion:
- The series converges for all real numbers.
- The radius of convergence is infinite.
- The interval of convergence is (-∞, +∞).