Consider the following series. sum_(n=0)^infinity (x+9)**n/(7**n) (a) Find the values of x for which the series converges. (Enter the smaller number first.) ( , ) (b) Find the sum of the series for those values of x.

Converges for x in (-16,2) and the value of convergence is

Step-by-step explanation:

Recall that a series of the form converges if |r|<1. The value of convergence, when the series converges is

The given series is

So, it converges if and only if

Equivalently, we get

Which is the interval . So the series converges over the interval (-16,-2).

The value of converge for an x in that interval is

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for this case we must apply distributive property to the following expression:

the distributive property states that:

then, when applying the property we have to multiply the -2 for each term within the parenthesis: