Question

Determine whether the power series converges?

If you know that the power series `sum_(n=0) ^oo c_n x^n` converges for `x=4` and diverges for `x=-6`, what can you conclude about `sum_(n=0) ^oo (84^n c_n)/(7^n)`?

Answer 1

Diverges.

Explanation:

If the series converges for `x=4` and diverges for `x=-6,` we may infer that it converges for all `x` in the interval `(-6, 4]` and diverges for all other `x.`

So, for the given power series, some simplification is needed to determine what exactly `x` is.

`sum_(n=0)^oo(c_n84^n)/7^n=sum_(n=0)^ooc_n(84/7)^n`

We see `x=84/7=12>4,` so the power series diverges.