Using d'Alembert method, R = what? #lim_(n->oo)# #((x+1)^n)/(2^(n)(n!))#

#lim_(n->oo)# #((x+1)^n)/(2^(n)(n!))#

1 Answer
Mar 29, 2018

The radius of convergence is #oo# (the series converges for all values of #x#)

Explanation:

We use the ratio test to find the radius of convergence.

#L = lim_(n -> oo) ((x + 1)^(n + 1)/(2^(n + 1)(n + 1)!))/((x + 1)^n/(2^n(n!)))#

#L = lim_(n-> oo) (x + 1)/(2(n + 1))#

#L = |x + 1| lim_(n -> oo) 1/(2n + 2)#

#L = |x + 1| * 0#

#L = 0#

Therefore this series converges absolutely (the radius of convergence is #oo#).

Hopefully this helps!