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Calc II problem. Sum of a series as a function of x?
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)?
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: \[\sum_{n=0}^\infty 9 {\frac{x-5}{9}}^n=\frac{9}{1-\frac{x-5}{9}}=\frac{81}{14-x}\]