Exponential sequences. The number e

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

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

State the limit of the following geometric sequence:

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

Calculate the limit of the following exponential sequence:

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

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.

A teacher proposes designing an exam exercise and seeks to construct a sequence of the form:\n$$a_n = \left(\frac{A n^2 + B n - 1}{3n^2 + 5}\right)^{C n}$$\nwhere $A$, $B$, and $C$ are constant integers ($A > 0$, $B \neq 0$, $C \neq 0$).\n\na) Determine with reasoning what numerical value the coefficient $A$ must necessarily have so that the sequence gives rise to an indeterminate form of type $1^\infty$. Explain what would happen to the limit if $A$ were greater or less than that value.\nb) With the value of $A$ fixed in the previous part, analytically deduce what mathematical relationship parameters $B$ and $C$ must satisfy so that the limit of the sequence is exactly equal to $\frac{1}{e^2}$.\nc) From the relationship found, provide two different triples of integers $(A, B, C)$ that work for this exercise. Select one of them and prove step by step that its limit is indeed $e^{-2}$.

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:

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

To calculate the limit $L = \lim_{n\to\infty} \left(\frac{n+4}{n-1}\right)^{2n+1}$, two classmates propose different strategies:\n- Aoife prefers Method A (fitting to the classical definition): algebraically manipulate the base by adding and subtracting terms to write it in the form $\left(1 + \frac{1}{f(n)}\right)$, and then compensate the exponent by multiplying and dividing by $f(n)$.\n- Seán prefers Method B (general formula for the indeterminate form $1^\infty$): verify that $\lim a_n = 1$ and $\lim b_n = \infty$, and calculate $e^{\lim b_n(a_n-1)}$ directly.\n\na) Fully develop the resolution of the limit using both methods and verify that both lead to the exact same result.\nb) Compare both approaches: which one requires more algebraic manipulations with fractions? In which one do you consider there is a greater risk of making mistakes with signs or exponents? Justify which of the two methods you would choose in an exam and why.

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:

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

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

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

An amount of $5000$ euros 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 $C_n = 5000 \times e^{0.04n}$. Calculate the accumulated capital after $10$ years and compare it with the capital obtained if the compounding were annual, whose expression is $C'_n = 5000 \times (1,04)^n$.

Aoife, who lives in Ireland, decides to deposit $12,000\text{ €}$ of her savings into a $1$-year fixed-term bank deposit. The bank offers a nominal annual interest rate of $4\%$ ($r = 0.04$) and allows choosing the frequency with which interest is compounded and reinvested. If the interest is compounded $n$ times a year, the accumulated final capital is given by the sequence $C_n = 12,000 \times \left(1 + \frac{0.04}{n}\right)^n$.\n\na) Model the final capital if the bank compounded interest instantaneously and continuously ($n \to \infty$). Express this exact value in terms of the number $e$ and calculate its amount rounded to the nearest euro cent (knowing that $e^{0.04} \approx 1.0408108$).\nb) Calculate the net profit in euros generated by the deposit with a single annual compounding ($n = 1$) and compare it with the profit obtained through continuous compounding.\nc) A commercial brochure states: "If we divide the year into increasingly shorter periods (seconds, milliseconds...), your interest will generate new interest continuously, making your profits grow without ceiling". Using the concept of limit and the properties of the number $e$, formally refute this advertising slogan and determine the maximum additional gain that continuous compounding offers compared to annual compounding.

An amount of $10000$ euros 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 $C_n = 10000 \left( 1 + \frac{0.05}{n} \right)^n$. 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 ($n=4$).

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 $24000$ euros, 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?

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 $n \to \infty$).