Question

b2-series: problem 6
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consider the following series. answer the following questions.
\\(\sum_{n = 1}^{\infty}\frac{x^{n}}{5^{n}}\\)

1. find the values of x for which the series converges.

answer (in help (intervals)):

1. find the sum of the series for those values of x. write the formula in terms of x.

sum: 1/(1 - x/5)
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Explanation:

Step1: Identify the series type

The series $\sum_{n = 1}^{\infty}\frac{x^{n}}{5^{n}}=\sum_{n = 1}^{\infty}(\frac{x}{5})^{n}$ is a geometric series with common - ratio $r=\frac{x}{5}$.

Step2: Find the convergence interval

For a geometric series $\sum_{n = 1}^{\infty}r^{n}$, it converges when $|r|\lt1$. So, for $r = \frac{x}{5}$, we have $|\frac{x}{5}|\lt1$. Solving the inequality $|\frac{x}{5}|\lt1$ gives $|x|\lt5$, or $- 5\lt x\lt5$.

Step3: Find the sum of the geometric series

The sum of an infinite geometric series $\sum_{n = 1}^{\infty}r^{n}$ with $|r|\lt1$ is given by $S=\frac{r}{1 - r}$. Here, $r=\frac{x}{5}$, so the sum $S=\frac{\frac{x}{5}}{1-\frac{x}{5}}=\frac{x}{5 - x}$ for $|x|\lt5$.

Answer:

1. $(-5,5)$
2. $\frac{x}{5 - x}$