Exponential Sequences. The Number e

Calculate the limit of the following exponential sequence:

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

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

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

Calculate the limit of the sequence with general term:

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

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

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

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 sequence:

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

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

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

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 second-degree polynomials in the base:

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

Calculate the limit of the following sequence, which features an algebraic exponent in fractional form:

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

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 limit of the sequence combining square roots in its base and a linear exponent:

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

Calculate the limit of the sequence by resolving the indeterminate form generated by the algebraic quotient in the exponent:

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

Find the value of the real parameter $a$ such that the following equality involving the number $e$ holds:

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

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

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

The concentration of a pollutant in a lake is naturally reduced by 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 infinitely many filtering stages are applied, prove mathematically to what exact fraction of the initial concentration $C_0$ the pollutant will be reduced.

An amount of $300\,000$ pesos is deposited in a financial institution at an annual interest rate of $4\%$. The formula for the final capital after $n$ years with continuous compounding is given by the sequence $C_n = 300\,000 \times e^{0.04n}$. Calculate the accumulated capital after $10$ years and compare it with the capital obtained if the compounding were annual, whose expression is $C'_n = 300\,000 \times (1,04)^n$.

An amount of $600\,000$ pesos is invested at a nominal annual interest rate of $5\%$. The sequence describing the final capital if interest is compounded $n$ times a year is $C_n = 600\,000 \left( 1 + \frac{0.05}{n} \right)^n$. Calculate the limit of this sequence as $n \to \infty$ to find the final capital under continuous compounding. Compare this result with the capital obtained if compounded quarterly ($n=4$).

State the limit of the following geometric sequence:

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

The amount of a radioactive isotope in a sample decreases exponentially according to the sequence $M_n = 200 \times e^{-0.1n}$, 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? What value does the mass tend to in the long term (as $n \to \infty$)?

A vehicle loses $15\%$ of its value each year due to depreciation. If its initial price at the dealership is $24000$ pesos, its value after $n$ years is given by the sequence $V_n = 24000 \times (0,85)^n$. What will its value be after $5$ years? What happens to its value as the number of years tends to infinity?

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

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