What is the interval of convergence of #sum {(x - 7)^n}/{(7)^n}#?

1 Answer
Nov 16, 2015

The interval of convergence for #sum(x-7)^n/7^n# is #(0, 14)#.

Explanation:

The ratio test states that a series #sum a_n#

  • Converges if #0 <= lim_(n->oo)|a_(n+1)/a_n| < 1#
  • Diverges if #lim_(n->oo)|a_(n+1)/a_n| > 1#
  • May or may not converge if #lim_(n->oo)|a_(n+1)/a_n| = 1#

We are looking for the interval on which #sum(x-7)^n/7^n# converges. Using the ratio test, we are thus looking for where
#lim_(n->oo)|(x-7)^(n+1)/7^(n+1)-:(x-7)^n/7^n| < 1#
#=> lim_(n->oo)|(x-7)/7| < 1#
#=> lim_(n->oo)|(x-7)| < 7#
#=> 0 < x < 14#

So we know that the interval of convergence includes #(0, 14)#. However, the ratio test does not tell us what happens at the endpoints, so we must look at those manually.

At #x=0# we have
#sum (-7)^n/7^n = sum(-1)^n# does not converge.

At #x = 14# we have
#sum 7^n/7^n = sum1# diverges.

Thus the interval of convergence for #sum(x-7)^n/7^n# is #(0, 14)#.