please help me...for the power series summation of n^4x^n/2(n-1)factorial....find the radius
i am guessing the radius is either x or x/2 but i am still not sure..CAN someone help me plz?
PLZ HELP ME...i don't know where to start
Take the ratio of two succesive terms and take the limit n --> infinity. For finite x the limit is zero, which means that the radius of convergence is infinity. In general, the radius of convergence is that value for x you have to substitute in the above limit to make it equal to 1. In this case the limit is zero for all x.
so the radius would be x right?
To find the radius of convergence for a power series, you can use the ratio test. The ratio test states that if
lim (n → ∞) |a_(n+1)/a_n| = L
where L is the limit, then the series converges if L < 1 and diverges if L > 1. If L = 1, the test is inconclusive.
For the given power series, n^4 * x^n / (2 * (n-1)!) , let's apply the ratio test:
|a_(n+1)/a_n| = [(n+1)^4 * x^(n+1) / (2 * n!) ] * [(2 * (n-1)!) / (n^4 * x^n)]
Simplifying this expression, we get:
|a_(n+1)/a_n| = [(n+1)^4 * x^(n+1) / (n^4 * x^n)] * [1 / (2 * n)]
Now, take the limit as n approaches infinity:
lim (n → ∞) |a_(n+1)/a_n| = lim (n → ∞) {[(n+1)^4 * x^(n+1) / (n^4 * x^n)] * [1 / (2 * n)]}
To find the limit, we can simplify the expression inside the limit:
lim (n → ∞) {[(n+1)^4 * x^(n+1) / (n^4 * x^n)] * [1 / (2 * n)]}
= lim (n → ∞) {(n+1)^4 * x / (n^4 * x^n)} * [1 / (2 * n)]
As n approaches infinity, the term containing n in the numerator becomes negligible compared to the term containing n^4 in the denominator. Hence, we can ignore it.
lim (n → ∞) {[(n+1)^4 * x / (n^4 * x^n)] * [1 / (2 * n)]}
= lim (n → ∞) {(n+1)^4 / (2 * n)}
Evaluating the limit, we get:
lim (n → ∞) {(n+1)^4 / (2 * n)}
= ∞ / (2 * ∞)
= 1/2
Since the limit is less than 1, the power series converges for all values of x. Therefore, the radius of convergence is infinite.
In this case, the radius of convergence is not determined by the value of x, and it is not x/2 as you mentioned. The power series for this function converges for all real values of x.