Consider the sequence $c_m=\frac{m-4}{2m+1}$. Does this sequence converge? If so, to what value?

Evaluate the limit of the sequence $\left\{b_n\right\}=2(0.4{)}^n+5(0.9{)}^n$ as $n\to \infty$, or determine if it diverges.

A patient receives a $100\text{ mg}$ dose of a medication every $12$ hours. The medication has a half-life of $12$ hours. Let $m_n$ be the amount of medication in the body after the $n$^th dose, with $m_1=100\text{ mg}$. Provide the recurrence relation that models this situation.

Simplify $\frac{n!}{(n+3)!}$ for any integer $n\ge 0$.

Find the value of $\frac{10,000!}{9,998!}$ exactly.

True or False: $k!k!k!=\left(3k\right)!$ for all positive integers $k$.

Imagine you set aside $d$ dollars at the start of each month into a savings account that offers a monthly interest rate of $i$, where $i$ is the annual interest rate divided by $12$ (for example, if the annual rate is $3.6\%$, then $i=\frac{0.036}{12}=0.003$). After the first month, your savings include your initial deposit plus the next deposit with interest applied, totaling $d+d(1+i)$. Extending this pattern, the total amount in your account after $n$ months is $S_n=d+d(1+i)+d(1+i{)}^2+\cdots +d(1+i{)}^{n-1}$. Find the total amount in your savings account after $5$ years ($60$ months) if you deposit $\$300$ each month and the monthly interest rate is $0.15\%$.

A factory produces widgets. Each month, production is $90\%$ of the previous month's output. In the first month, $2000$ widgets are produced. What is the greatest total number of widgets that can be produced over time?

A person is given $100\text{ mg}$ of a drug every $8$ hours. The half-life of the drug is $8$ hours. What is the steady-state amount of the drug in the person's body, using infinite series?