Question

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

Answer 1

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)`.