How do you find the radius of convergence Sigma (1*4*7* * * (3n+1))/(n!)x^n from n=[0,oo)?

1 Answer
Jan 17, 2017

The radius of convergence is R= 1/3

Explanation:

The ratio test states that a necessary condition for sum_(n=0)^oo a_n to converge is that:

L = lim_(n->oo) abs(a_(n+1)/a_n) <= 1

if L<1 the condition is also sufficient and the series converges absolutely, while for L=1 the test is inconclusive.

We then evaluate the ratio for the series at hand:

abs(a_(n+1)/a_n) = (abs(x)^(n+1)(prod_1^(n+1) (3k+1))/((n+1)!))/(abs(x)^n(prod_1^n (3k+1))/(n!))= abs(x)(3(n+1) +1)/(n+1) = abs(x) (3n+4)/(n+1)

so that:

lim_(n->oo) abs(a_(n+1)/a_n) = 3abs(x)

The series is then absolutely convergent for absx <1/3 and divergent for absx >1/3with means that the radius of convergence is R= 1/3