Exercises: Limit of a sequence

Determine whether the sequence whose general term is $a_n = \frac{(-1)^n \times 5}{n}$ is convergent, divergent or oscillating. If it is convergent, state its limit.

Calculate the limit of the following polynomial sequence as $n$ tends to infinity:

$b_n = -2n^3 + 7n^2 - 4n + 12$

Find the limit of the rational sequence given by the expression:

$c_n = \frac{3n^2 - 5n + 1}{8n^3 + 2n^2 - n}$

Calculate the limit of the following sequence, first identifying the degrees of the numerator and denominator:

$d_n = \frac{6n^2 - 3n + 5}{2n^2 + 4n - 1}$

Determine the value that the sequence $e_n$ tends to for very large values of $n$:

$e_n = \frac{n^4 + 2n^2 - 7}{5n^2 - 3n + 1}$

Study the behaviour and calculate the limit of the exponential-type sequence:

$f_n = 12 \times \left(\frac{3}{4}\right)^n$

Applying the definition of the number $e$, calculate the limit of the following sequence:

$g_n = \left(1 + \frac{1}{n}\right)^{7n}$

Calculate the limit of the sequence involving a simple square root in the numerator:

$h_n = \frac{\sqrt{16n^2 + 3n}}{2n + 5}$

Lucas is solving the limit of the sequence $a_n = \dfrac{3n + (-1)^n \times 4}{2n - 1}$ and writes on his exam: "Since the term $(-1)^n$ alternates indefinitely between $-1$ and $1$, the sequence jumps around all the time and, therefore, it is an oscillating sequence that has no limit". Identify Lucas's conceptual error, explain why his statement is false, and correctly calculate $\displaystyle\lim_{n \to \infty} a_n$, classifying the sequence as convergent, divergent, or oscillating.

Consider the sequence with general term $a_n = \dfrac{4n^2 - 5}{2n^2 + 3n}$. Before operating formally, calculate the value of the terms to two decimal places for $n = 10$, $n = 100$, and $n = 1000$, and make a reasoned estimate from them of the value you think the sequence approaches. Next, calculate the limit analytically by comparing polynomial degrees and check whether your initial estimate was correct, stating whether the sequence is convergent or divergent.

The temperature in degrees Celsius of a piece of metal cooling in a room follows the sequence $T_n = 22 + 85 \times (0,82)^n$, where $n$ is the time elapsed in minutes. Calculate the limit of the sequence $T_n$ as $n \to \infty$ and interpret the result in the context of the problem.

Resolve the indeterminate form that arises when calculating the limit of the following difference of rational sequences:

$\lim_{n \to \infty} \left( \frac{n^2 + 5n}{n + 2} - n \right)$

Calculate the limit of the following irrational sequence, multiplying and dividing by the conjugate expression to resolve the indeterminate form:

$\lim_{n \to \infty} \left( \sqrt{n^2 + 6n} - n \right)$

The fish population of a sustainable fish farm grows according to the mathematical model $P_n = \frac{150000n}{3n + 250}$, where $n$ represents the number of days since the introduction of the fingerlings. Determine the carrying capacity of the environment, that is, the long-term limit of the fish population.

Calculate the limit of the sequence, transforming the expression to apply the limit definition of the number $e$:

$\lim_{n \to \infty} \left( \frac{n + 7}{n + 2} \right)^{4n}$

An amount of $5000$ euros is deposited into an investment fund offering an annual interest rate of $3.5\%$. If the institution compounds the interest $n$ times a year, the capital accumulated after $10$ years is given by the sequence $C_n = 5000 \times \left(1 + \frac{0.035}{n}\right)^{10n}$. Calculate the limiting capital that would be obtained if compounding were continuous ($n \to \infty$).

Find the limit of the following sequence, first expanding the algebraic identities in the numerator to identify the terms of highest degree:

$\lim_{n \to \infty} \frac{(3n+2)^2 - (3n-2)^2}{n+5}$

Starting from a square of area $A_1 = 400 \text{ cm}^2$, a sequence of squares is constructed by joining the midpoints of the sides of the previous square, as shown in the figure. Given that the area of each new square is half the area of the previous square, calculate the limit of the sum of the areas of the infinitely many squares thus constructed: $S_\infty = \lim_{n \to \infty} (A_1 + A_2 + \dots + A_n)$.

The Central Statistics Office (CSO) projects the population trend of a small rural village in County Kerry. According to this model, the population $P_n$ (expressed in hundreds of inhabitants) after $n$ decades elapsed since the year 2020 is given by the sequence $P_n = \dfrac{18n + 45}{3n + 5}$. Calculate the estimated initial population in 2020 ($n = 0$), the one projected for 2040 ($n = 2$), and the limit $\displaystyle\lim_{n \to \infty} P_n$. From this last value, interpret in the context of the problem whether the model predicts the disappearance of the village due to depopulation or its long-term stabilization, indicating the exact number of inhabitants it would tend towards.

Design the general term of a rational sequence $a_n = \dfrac{P(n)}{Q(n)}$ (where $P(n)$ and $Q(n)$ are degree 1 polynomials with integer coefficients) that simultaneously satisfies three conditions: its first term must be $a_1 = 2$, it must be strictly increasing for all $n \geq 1$, and its limit as $n \to \infty$ must be exactly $5$. Once your formula for $a_n$ is found, calculate the natural number $N$ from which it is verified that the distance between the terms of the sequence and its limit is less than $\varepsilon = 0.02$ (that is, $|a_n - 5| < 0.02$ for all $n > N$).

Find the values of the real parameters $a$ and $b$ (with $b > 0$) such that the following equality holds:

$\lim_{n \to \infty} \left( \sqrt{9n^2 + an + 2} - bn \right) = 4$

Given the sequence $a_n = \frac{4n - 3}{n + 1}$, we know that its limit is $L = 4$. Using the formal definition of a limit, find from which term $n$ onwards the distance between the terms of the sequence and its limit is strictly less than one thousandth, that is, $|a_n - 4| < 0.001$.

Solve the following limit that leads to an indeterminate form of the type $1^\infty$, paying special attention to the algebraic operations in the exponent:

$\lim_{n \to \infty} \left( \frac{n^2 - n + 3}{n^2 + n + 1} \right)^{n^2 - 1}$

The figure shows the first three iterations in the construction of the Koch curve. If we start from an initial segment of length $L_0 = 1$ metre (panel A), at each step $n$ the total length of the curve is multiplied by $\frac{4}{3}$ compared to the previous step. Write the expression for the length $L_n$ at step $n$ and calculate $\lim_{n \to \infty} L_n$. What geometric conclusion can be drawn from this result?

Calculate the limit of the following sequence, which features a double square root and requires careful application of the conjugate technique:

$\lim_{n \to \infty} n \times \left( \sqrt{n^2 + 4} - n \right)$

A patient receives a daily dose of a medication. After absorption and elimination throughout the day, the concentration of the drug in the blood just before the next dose follows the recurrence relation $C_n = 15 + C_{n-1} \times 0.4$, where $15 \text{ mg/L}$ is the contribution of the new dose and the factor $0.4$ indicates that $40\%$ of the previous day's drug remains. If the initial concentration is $C_1 = 15$, calculate the long-term equilibrium concentration in the blood, that is, $\lim_{n \to \infty} C_n$.

A sequence is defined by the following recurrence relation: the first term is $a_1 = \sqrt{6}$, and each subsequent term is obtained as $a_{n+1} = \sqrt{6 + a_n}$. Assuming that the sequence is increasing and bounded above (and is therefore convergent and has a limit $L$), set up the equation that the limit must satisfy and find it.

A high-tech elastic cable undergoes iterative tension tests. Its initial length at rest is $20$ metres. In the first test it stretches $4$ metres. In each successive test, the material loses elasticity and stretches by $75\%$ of the amount it stretched in the immediately preceding test. The total length after $n$ tests is given by $L_n = 20 + \sum_{k=1}^n 4 \times (0,75)^{k-1}$. Calculate the theoretical maximum length that the cable will reach if subjected to infinitely many tests.

To calculate the limit of the sequence $a_n = \sqrt{4n^2 + 7n} - 2n$, two students discuss which strategy to follow. Candela suggests a shortcut: "Since $7n$ grows much slower than $4n^2$, we can neglect it inside the square root; thus we have $\sqrt{4n^2} - 2n = 2n - 2n = 0$, so the limit is $0$". Daniel argues that this is not valid because it is an indeterminate form $\infty - \infty$ and one must multiply and divide by the conjugate. Solve the limit by applying Daniel's rigorous method, explain why Candela's shortcut leads to an incorrect result, and analyze what would happen if the sequence were $b_n = \sqrt{4n^2 + 7n} - 3n$: is the conjugate still necessary in this second case, or is there a direct deduction based on orders of magnitude?

Given the sequence $a_n = \dfrac{3n - 1}{n + 2}$: first calculate $\displaystyle\lim_{n \to \infty} a_n$ and formally prove this result by applying the rigorous definition of limit, deducing a formula that provides the minimum value of $N \in \mathbb{N}$ as a function of any margin $\varepsilon > 0$. Use your formula to calculate $N$ when $\varepsilon = 0.001$. Finally, a classmate observes that for $n = 100$ the distance between consecutive terms is already very small ($|a_{101} - a_{100}| < 0.001$) and deduces that, therefore, $a_{100}$ must already be within less than $0.001$ of its limit. Check numerically whether their deduction is true or false for this term and explain the conceptual difference between the closeness of two consecutive terms and the closeness of a term to its limit.