Express $0.5272727\dots$ as a ratio of two integers.

Determine if the series $\sum _{m=1}^{\infty }\frac{5m}{4m-2}$ diverges using the Divergence Test, or state if the test is inconclusive.

You take $60\text{ mg}$ of a vitamin supplement daily. Each day, $60\%$ of the vitamin is eliminated from your body. Let $v_n$ be the amount of the vitamin in your body after the $n$^th dose, with $v_1=60$. What is the steady-state amount of the vitamin in your body? Solve the problem by writing the amount of the vitamin in your body as a geometric series and evaluating the series.

Find the sum of the infinite series $\sum _{j=1}^{\infty }(\frac{6}{5^j}-\frac{6}{5^{j+1}})$ using a telescoping series argument.

Determine the sum of the geometric series $\sum _{m=0}^{\infty }{\left(\frac{2}{3}\right)}^m$ or state that the series diverges.