Question

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{1}{5^{k}-1}$$

Short answer

Explain. Answer: The series converges. By applying the Ratio Test and computing the limit of the ratio of consecutive terms, we found that the limit is $$\frac{1}{5}<1$$. Since the limit of the ratio is less than 1, the series converges according to the Ratio Test.

Step-by-step solution

Set up the Ratio Test

Let's denote the series terms as \(a_k = \frac{1}{5^k - 1}\). To apply the Ratio Test, we need to calculate the limit of the ratio of consecutive terms, that is, $$\lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right|$$

Calculate the Ratio

Substitute the definition of \(a_k\) into the ratio. It becomes: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_{k}} = \lim_{k \to \infty} \left|\frac{\frac{1}{5^{k+1}-1}}{\frac{1}{5^k-1}}\right| = \lim_{k \to \infty} \left|\frac{5^k-1}{5^{k+1}-1}\right|$$

Simplify the Ratio

Factor out a \(5^k\) from both the numerator and the denominator to simplify the expression: $$\lim_{k \to \infty} \left|\frac{5^k(1-\frac{1}{5^k})}{5^k(5-\frac{1}{5^k})}\right| = \lim_{k \to \infty} \left|\frac{1-\frac{1}{5^k}}{5-\frac{1}{5^k}}\right|$$

Calculate the Limit

As \(k\) goes to infinity, the term \(\frac{1}{5^k}\) approaches zero. Thus, the limit becomes: $$\lim_{k \to \infty} \left|\frac{1-\frac{1}{5^k}}{5-\frac{1}{5^k}}\right| = \left|\frac{1}{5}\right|$$

Test Convergence

Now that we have found the limit of the ratio, we can apply the Ratio Test to determine the convergence of the series: - The limit is $$\frac{1}{5}<1$$, so the series $$\sum_{k=1}^{\infty} \frac{1}{5^{k}-1}$$ converges.