Power Series question, i am stuck at a point.?

Question

ok, the given series is, summation 1 to infinity of (n^4 * x^n)/2(n-1)! and we need to find the radius and interval of convergence

the ! is the factorial symbol not me exclaiming... hehe...
ok so
an + 1/ an = absval.( (n+1)^4 * (x^n+1) / 2(n)! * 2(n-1)!/n^4 x^n )

absval. ( (n-1) (n + 1)^4 x / n^4 )
(1-1/n)(1+1/n) / 1/n (x)

and then what? what am i supposed to do with 1/n in the denominator.

Answer 1

I dodn't understand what you did. Anyway, for every n,

|a(n+1)/a(n)| =| [(n+1)^4]/(n^4) * [(2(n-1!])/(2 n!) * x^(n+1)/x^n | = (1 + 1/n)^4 * 1/n * |x|

n --> oo => (1 + 1/n)^4 --> 1 and 1/n --> 0,so that lim |a(n+1)/a(n)| = 0 for any real x. (even for any complex x).

So, your series converges for every real (and even complex) x. Its convergenc e radius is oo and the interval of convergence is (-oo, oo) that is, the whole R.