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

1 Answer
sente
Nov 16, 2015

The interval of convergence for sum(x-7)^n/7^n∑(x−7)n7n is (0, 14)(0,14).

Explanation:

The ratio test states that a series sum a_n∑an

  • 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).

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