Exercises: Exponential sequences. The number e.

Calculate the limit of the following exponential sequence:

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

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

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

State the limit of the following geometric sequence:

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

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, given 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 sequence:

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

Seán solves the following limit of a sequence on the blackboard: $\lim_{n\to\infty} \left(\frac{n+5}{n+2}\right)^{3n}$. To justify his result, he writes in his notebook: "Since $\lim_{n\to\infty} \frac{n+5}{n+2} = 1$ and the number $1$ raised to any power always gives $1$, the limit is simply $1^{3n} = 1$".\n\na) Identify the conceptual error that Seán makes by treating the expression as if it were not an indeterminate form. Explain why in the limit process the form $1^\infty$ does not automatically equal the number $1$.\nb) Correct the exercise step by step using the formula $\lim a_n^{b_n} = e^{\lim b_n(a_n-1)}$, determining the true value of the limit as a power of the number $e$.

Consider the sequence $a_n = \left(1 - \frac{1}{2n}\right)^n$.

1. a) Estimate: calculate exactly the first two terms, $a_1$ and $a_2$, and express them in decimal form. Also calculate, with a calculator, $a_{10}$ and $a_{100}$. Observe whether these values are less or greater than $1$ and how they evolve. From what you observe, reason intuitively: when raising a base less than $1$ to an exponent that grows towards $+\infty$, the base also changes and gets closer to $1$. What do you think happens with the limit? Give an estimate of its value and justify it without applying limit formulas yet.

1. b) Check: calculate the limit $\lim_{n\to\infty} a_n$ by applying the formula for the number $e$, that is, $\lim_{n\to\infty}\left(1+\frac{1}{k_n}\right)^{k_n}=e$ when $k_n\to\infty$ (or the equivalent version for the indeterminate form $1^\infty$). Express the result in the most simplified form possible and compare it with your estimate from part a).

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

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

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 $C'_n = 5000 \times (1,04)^n$.

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

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

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 $n \to \infty$)?

Find the limit of the sequence combining second-degree polynomials in the base:

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

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?

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

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

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 $n \to \infty$.

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.

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.

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

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

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 $n \to \infty$).

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:

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

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

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

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.

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

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

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 ($n=4$).

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 $e^{-2}$.

Consider the following family of exponential sequences depending on a natural parameter $k \in \{1, 2, 3, 4, \ldots\}$:\n$$s_n = \left(\frac{n^k + 4n}{n^k + 1}\right)^{n^3}$$\n\na) Estimate and analyze the growth rates: if $k = 1$, does the indeterminate form $1^\infty$ appear? If $k \ge 2$, the base tends to $1$ and the exponent to $+\infty$. Analyze the degrees of the numerator and denominator when performing the subtraction $(a_n - 1)$, and deduce for what critical value of $k$ the rate at which the base approaches $1$ will exactly balance the rate at which the exponent $n^3$ grows towards $+\infty$, so that the limit is a finite power of $e$ different from $1$ and from $+\infty$.\nb) Formally verify your deduction by calculating the limit $\lim_{n\to\infty} s_n$ in the following three cases: for $k = 2$, for the critical value of $k$ you predicted, and for $k = 5$.\nc) In light of the results, state a reasoned general conclusion about the value of $\lim_{n\to\infty} s_n$ depending on whether $k$ is less than, equal to, or greater than that critical value.

Topic Challenge. How does capital evolve under compound interest, and how can we represent it dynamically? Design a financial and geometric model using real data.

1. Real data: Go to the Banco de España website and look up the current Euribor rate ($r$, e.g., $3.2\,\% \rightarrow r = 0.032$). Set an initial capital of $1000$ €.

2. Modeling and the number $e$: Formulate the general term of the accumulated capital according to the compounding periods: $C_n = 1000\left(1 + \frac{r}{n}\right)^n$. Check in a spreadsheet whether it is an increasing or decreasing sequence, justify whether it is bounded, and calculate its limit as $n \to \infty$ using the number $e$. Contrast this exponential model with an arithmetic progression of constant monthly savings.

3. Dynamics in the complex plane: Model the quarterly reinvestment in GeoGebra by assigning a complex number in polar form to each stage: $z_k = (C_k)_{k \times \frac{\pi}{4}}$. Successively multiply by $w = \left(1 + \frac{r}{4}\right)_{\pi/4}$ to generate a geometric progression of complex numbers. Justify using operations in polar form whether the resulting spiral is increasing or decreasing.

4. Final report: Write a reasoned proposal comparing both models with their table and graph.

Assessment criteria:

• Rigour in obtaining real data and formulating the general term.

• Analytical justification of monotonicity, boundedness, and calculation of the limit using the number $e$.

• Correctness in operations with complex numbers in polar form and their representation in GeoGebra.