Question

Calc II problem. Sum of a series as a function of x?

Answer 1

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. \[\sum_{n=0}^{\infty} 9((x-5)/9)^n\]
series converges from -4 to 14. But how do I express the sum as a function of F(x)?

Answer 2

As you might know, the following is true:
\[\sum_{n=0}^\infty x^n=\frac{1}{1-x}\]
You can use that here: let's define y:
\[y=\frac{x-5}{9}\]
we get:
\[\sum_{n=0}^\infty 9y^n = \frac{9}{1-y}\]
and so:
\[\sum_{n=0}^\infty 9 {\frac{x-5}{9}}^n=\frac{9}{1-\frac{x-5}{9}}=\frac{81}{14-x}\]