step 3 now, our goal is to find a power series for - 1 / (6 - x) and then integrate it. factor -1 from the…

step 3 now, our goal is to find a power series for - 1 / (6 - x) and then integrate it. factor -1 from the numerator and 6 from the denominator. this will give us - 1 / (6 - x)=-1/6(1 / (1 - (x / 6))). step 4 therefore, we get - 1 / (6 - x)=-1/6 sum(n = 0 to infinity) ( )^n. submit skip (you cannot come back) need help? read it submit answer

step 3 now, our goal is to find a power series for - 1 / (6 - x) and then integrate it. factor -1 from the numerator and 6 from the denominator. this will give us - 1 / (6 - x)=-1/6(1 / (1 - (x / 6))). step 4 therefore, we get - 1 / (6 - x)=-1/6 sum(n = 0 to infinity) ( )^n. submit skip (you cannot come back) need help? read it submit answer

Answer

Explanation:

Step1: Recall geometric - series formula

The geometric - series formula is $\frac{1}{1 - t}=\sum_{n = 0}^{\infty}t^{n}$, for $|t|\lt1$. In Step 3, we have $-\frac{1}{6 - x}=-\frac{1}{6}\frac{1}{1-\frac{x}{6}}$. Here, $t=\frac{x}{6}$.

Step2: Substitute into geometric - series formula

Substituting $t = \frac{x}{6}$ into the geometric - series formula $\frac{1}{1 - t}=\sum_{n = 0}^{\infty}t^{n}$, we get $-\frac{1}{6 - x}=-\frac{1}{6}\sum_{n = 0}^{\infty}(\frac{x}{6})^{n}$.

Answer:

$\frac{x}{6}$