Find the interval of convergence for the given power series...?

Question

Sum from Inf. to n = 1 of:

((x-10)^n)/(n*((-7)^n))

The series is convergent,

Left end limit, x = ____ (Included or not) Y/N
Right end limit, x = ____ (Included or not) Y/N

I think I already solved it, but I only have one more try, so I would like to compare to someones elses results. Mine were...

{3,17] , where { is not inclusive, and ] is inclusive.

Thanks for any help.

Answer 1

The ratio test gives:
{[(x-10)^(n+1)]/[(n+1)(-7)^(n+1)]}[(n)(-7)^n]/(x-10)^n
= (x-10)n/[(n+1)(-7)= (x-10)/(-7-7/n) goes to (x-10)/-7 when
x goes to infinity.

Thus the interval of convergence is |(x-10)/-7| < 1
or -1 < (x-10)/7 < 1
-7 3 < x < 17
When =3 we get the harmonic serries which is divergent so 3 is not included.. When we put x =17 we get the negative of the harmonic series so 17 is not included.

You are right on the money.