Exponential Sequences and the Number e

Find the limit of the sequence formed by the quotient of polynomials raised to $n$:

$c_n = \left( \frac{n+1}{n} \right)^n$

Determine the value to which the sequence $b_n$ tends as $n \to \infty$:

$b_n = \left( 1 + \frac{1}{2n} \right)^{2n}$

State the limit of the following geometric sequence:

$d_n = 5 \cdot \left( \frac{2}{3} \right)^n$

A vehicle loses $15\%$ of its value each year due to depreciation. If its initial price at the dealership is $\$24000$, the value after $n$ years is given by the sequence $V_n = 24000 \cdot (0.85)^n$. What will its value be after $5$ years? What happens to its value as the number of years tends to infinity?

A patient receives a dose of $50$ mg of a medication every $24$ hours. Knowing that the body eliminates $40\%$ of the medication present in that time interval, the accumulated amount in the body just after the $n$-th dose follows the geometric sequence $C_n = 50 + 50 \cdot 0.6 + 50 \cdot 0.6^2 + \dots + 50 \cdot 0.6^{n-1}$. Calculate the long-term accumulated amount (the limit of the sequence when $n \to \infty$).

Calculate the limit of the following exponential sequence:

$a_n = \left( 1 + \frac{1}{n} \right)^{3n}$

Calculate the limit of the sequence with general term:

$e_n = \left( 1 - \frac{1}{n} \right)^n$

Determine the limit of the following sequence, knowing that it presents an indeterminate form of the type $1^\infty$:

$f_n = \left( 1 + \frac{4}{n} \right)^n$

Find the limit of the exponential sequence whose exponent is $n+5$:

$g_n = \left( 1 + \frac{1}{n} \right)^{n+5}$

Calculate the limit of the following sequence, resolving the corresponding indeterminate form:

$a_n = \left( \frac{n+3}{n-1} \right)^{2n}$

Determine the limit of the following sequence by resolving the indeterminate form of the type $1^\infty$:

$b_n = \left( \frac{2n+5}{2n+1} \right)^{3n-2}$

Find the limit of the sequence combining quadratic polynomials in the base:

$c_n = \left( \frac{n^2+2n}{n^2-1} \right)^{5n}$

Calculate the limit of the following sequence, which has an algebraic exponent in the form of a fraction:

$d_n = \left( \frac{n-2}{n+4} \right)^{\frac{n^2+1}{n}}$

Calculate the limit of the exponential sequence formed by second-degree polynomials in both the base and the exponent:

$a_n = \left( \frac{2n^2-3n+1}{2n^2+n-5} \right)^{\frac{n^2+3}{n}}$

Find the value of the real parameter $a$ so that the following equality with the number $e$ is satisfied:

$\lim_{n \to \infty} \left( \frac{n+a}{n+1} \right)^{2n} = e^6$

Calculate the value of the parameters $a$ and $b$ so that the following sequence has a limit of $e^{-4}$, knowing that $a$ is a positive real number:

$\lim_{n \to \infty} \left( \frac{an+3}{2n+1} \right)^{bn} = e^{-4}$

Find the limit of the sequence that combines square roots in its base and a linear exponent:

$c_n = \left( \sqrt{\frac{n+2}{n-1}} \right)^{3n+1}$

The amount of a radioactive isotope in a sample decreases exponentially according to the sequence $M_n = 200 \cdot e^{-0.1n}$, where $n$ is the number of years elapsed and $M_n$ is measured in grams. How many grams will remain in the sample after $20$ years? To what value does the mass tend in the long term (as $n \to \infty$)?

In a thermodynamics experiment, the temperature difference between a metallic body and its surroundings decreases exponentially following the sequence $T_n = 80 \cdot e^{-0.05n}$, where $n$ is the minutes elapsed since the start of cooling. Calculate the temperature difference at $30$ minutes and determine the limit of the sequence when $n \to \infty$.

The concentration of a pollutant in a lake is naturally reduced through a process that follows the sequence $C_n = C_0 \left( 1 - \frac{2}{n} \right)^n$, where $n$ represents the number of filtering stages applied. If infinite filtering stages are applied, mathematically prove to what exact fraction of the initial concentration $C_0$ the pollutant will be reduced.

$\$5000$ are deposited in a financial institution at an annual interest rate of $4\%$. The formula for the final balance after $n$ years with continuous compounding is given by the sequence $C_n = 5000 \cdot e^{0.04n}$. Calculate the accumulated balance after $10$ years and compare it with the balance obtained if the compounding were annual, whose expression is $C'_n = 5000 \cdot (1.04)^n$.

$\$10000$ are invested at a nominal annual interest rate of $5\%$. The sequence describing the final balance if the interest is compounded $n$ times a year is $C_n = 10000 \left( 1 + \frac{0.05}{n} \right)^n$. Calculate the limit of this sequence when $n \to \infty$ to find the final balance under continuous compounding. Compare this result with the balance obtained if compounded quarterly ($n=4$).