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

Evaluate the infinite series: $\sum _{m=0}^{\infty }\left({\tan }^{-1}\right(m+3)-{\tan }^{-1}(m+1\left)\right)$.

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.

Estimate the value of the convergent series $\sum _{x=1}^{\infty }\frac{{(-1)}^x}{x^3}$ with an absolute error less than $2\times {10}^{-3}$. Round the answer to three decimal places.

Teams X and Y enter a sudden-death shootout after tying in a game. The teams take turns attempting shots, and the first team to score wins the shootout. Each team has a $\frac{1}{5}$ chance of scoring on their attempt, and Team X shoots first. The probability that Team X eventually wins can be expressed as the series $\sum _{k=0}^{\infty }\frac{1}{5}{\left(\frac{4}{5}\right)}^{2k}$. Find the value of this series.