Exercises: Sequences; limit of a sequence

Given the sequence whose general term is $a_n = \frac{1}{n^2}$, with $n = 1, 2, 3, \dots$

1. a) Write the first four terms of the sequence.

1. b) Reason intuitively what the value of its limit is as $n$ tends to infinity:

$$\lim_{n \to \infty} a_n$$

Calculate the limit of the following rational sequence by comparing the degrees of the numerator and the denominator:

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

Determine the value of the limit of the sequence $b_n$ as $n$ grows indefinitely:

$$b_n = \frac{100n + 50}{n^2 - 1}$$

On a social network, the number of "likes" on a post during the first few hours follows the sequence $c_n = \frac{4n^2}{n+2}$, where $n$ is the number of hours since publication. What happens to the number of "likes" in the very long term? Calculate:

$$\lim_{n \to \infty} \frac{4n^2}{n+2}$$

Calculate the limit of the following sequence involving a square root, by analyzing the highest-degree term:

$$d_n = \frac{\sqrt{9n^2 + n}}{2n - 3}$$

The graph shows some of its first terms (starting from $n=3$).

Using the comparison of orders of infinity (exponential vs polynomial), find the limit of the sequence:

$$e_n = \frac{3^n}{n^{10} + 500}$$

It is estimated that the value of a laptop computer after $n$ years is given by the sequence $v_n = 1200 \cdot (0,8)^n$. Calculate the limit of this sequence and interpret what the result means in the context of the device's depreciation.

$$\lim_{n \to \infty} 1200 \cdot (0,8)^n$$

The figure shows, for $n=1,2,\dots,10$, the terms of two sequences: the blue points correspond to $a_n=\left(1+\frac{1}{n}\right)^{n}$ and the green points to

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

1. a) Identify which of the two sequences in the figure (blue or green) corresponds to the definition of the number $e$ and write its limit.

1. b) Calculate $\displaystyle\lim_{n\to\infty} x_n$ (the green points), reasoning from the definition of $e$, and check that the result is consistent with the trend observed in the figure.

In a Mathematics II exam, Marcos has to calculate the limit of the following sequence:

$$\lim_{n \to \infty} \frac{5^n + n!}{2\cdot n! - 3^n}$$

To solve it, Marcos writes the following reasoning:

"In the numerator, $5^n$ is an exponential function with base 5, so it dominates over $n!$. In the denominator, the same happens with $3^n$, which dominates over $2\cdot n!$. Therefore, the limit is equivalent to $\lim_{n \to \infty} \frac{5^n}{-3^n} = -\lim_{n \to \infty} \left(\frac{5}{3}\right)^n = -\infty$".

1. a) Identify and explain in detail the conceptual error Marcos makes regarding the hierarchy of infinities ($\log n \ll n^k \ll a^n \ll n! \ll n^n$).
2. b) Correct his solution: factor out the term that truly dominates at infinity in both the numerator and the denominator, and calculate the correct value of the limit.

Given the sequence with general term:

$$a_n = \frac{6n^2 + \cos(n)\cdot \sqrt{n}}{2n^2 + 5}$$

1. a) Estimate: Analyze which terms grow fastest in the numerator and in the denominator as $n$ tends to infinity. Taking into account that the values of the cosine function oscillate boundedly in the interval $[-1, 1]$, make an intuitive and reasoned estimation of the value to which the sequence must converge before carrying out formal calculations.
2. b) Verify: Formally prove your estimation by applying the squeeze theorem. To do so, construct two simple sequences $b_n$ and $c_n$ that satisfy the inequality $b_n \le a_n \le c_n$ for all $n \ge 1$, calculate $\lim_{n \to \infty} b_n$ and $\lim_{n \to \infty} c_n$, and confirm whether the obtained result matches the estimated value.

Calculate the limit of the following sequence by analysing the degrees of the polynomials and justifying the result by comparing the leading terms:

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

Given the following sequences, use the hierarchy of infinities to determine, without performing extensive algebraic operations, the value of their limit as $n \to \infty$:

1. a) $b_n = \frac{n^5 + 10^6}{\ln(n^2 + 1) + 0,5^n}$

1. b) $c_n = \frac{\sqrt{n^3 + 4n}}{2n + \sqrt[3]{n^5}}$

1. c) $d_n = \frac{1,02^n}{n^{1000} + 5n}$

Solve the limit of the sequence $a_n = \sqrt{4n^2 + 3n - 1} - 2n$.

To visualize its behavior, the figure shows the first eight terms $a_1, a_2, \dots, a_8$. Observe which value they seem to approach and then justify the result by calculating the limit (for example, by multiplying and dividing by the conjugate).

An investor analyses the growth of a cryptocurrency whose daily evolution follows an exponential sequence rule. The relative value after $n$ days is given by:

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

Calculate the value to which the currency tends in the long term using the relation with the number $e$.

Find the values of the real parameters $k$ and $t$ such that the following equality holds:

$\lim_{n \to \infty} \left( \frac{kn + 5}{3n - 1} \right)^n = e^t$

Given that the limit of the base is $1$ and the final result is $e^2$.

Study the behavior of the recurrence sequence defined by:

$a_1 = 2$

$a_{n+1} = \frac{1}{2} \left( a_n + \frac{2}{a_n} \right)$

If we assume that the sequence is convergent and that its limit is $L$ ($L > 0$), calculate this value by solving the equation $L = \frac{1}{2} \left( L + \frac{2}{L} \right)$. Which famous irrational constant does this result correspond to?

In a biotechnology laboratory, the number of bacteria of a strain (in millions) after $n$ hours is modelled by the sequence:

$P_n = \frac{12n^2 + \text{sen}(n)}{3n^2 + 500}$

Determine the bacterial population as time tends to infinity. Explain what happens to the term $\text{sen}(n)$ in the limit calculation.

Calculate the limit of the following sequence, which involves a subtraction of fractions with different denominators:

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

State whether the sequence is divergent or convergent after resolving the indeterminate form $\infty - \infty$.

The National Statistics Institute (INE) studies the demographic evolution of a rural municipality in the region of Molina de Aragón (Guadalajara). The number of inhabitants at the beginning of year $n+1$, denoted by $P_{n+1}$, is modeled from the population of the previous year $P_n$ using the following recurrence relation:

$$P_{n+1} = 0,95\cdot P_n + 30$$

The coefficient $0,95$ models a net annual loss of $5\,\%$ due to natural increase and net migration, while the constant $30$ represents the new residents who register each year attracted by incentives for teleworking.

1. a) If in the initial year the census is $P_1 = 800$ inhabitants, calculate with justification the population of the following three years ($P_2$, $P_3$, and $P_4$) rounding the results to integers, and indicate the immediate trend you observe.
2. b) Knowing that the sequence is convergent and that its finite limit $L = \lim_{n \to \infty} P_n$ satisfies the fixed-point equation $L = f(L)$, calculate the number of inhabitants the municipality's population will tend to in the long term.
3. c) Interpret the value of $L$ in the sociodemographic context of the problem: will the town end up completely depopulated or will it reach a stable demographic equilibrium? Justify your answer.

Consider the sequence given by:

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

where $a$ and $b$ are fixed real constants, with $b > 0$.

1. a) Justify that the base of the power tends to $1$ and the exponent to $+\infty$ as $n \to \infty$, identifying the type of indeterminate form presented by the sequence.
2. b) Inverse problem: Determine the exact values of the parameters $a$ and $b$ knowing that the sequence converges to the real number $e^4$ and that both satisfy the linear relation $a - 2b = 1$. Show the entire procedure applying the equivalence $\lim_{n \to \infty} (u_n)^{v_n} = e^L$, with $L = \lim_{n \to \infty} v_n \cdot (u_n - 1)$.
3. c) Using the values found, calculate with reasoning the limit of the auxiliary sequence $s_n = \ln(a_n)$, justifying the limit and continuity property you use.

Calculate the value of the following limit of a sequence which yields an indeterminate form of the type $1^\infty$, paying special attention to the algebraic manipulation of the base and the exponent:

$$\lim_{n \to \infty} \left( \frac{2n^2 + 5n - 1}{2n^2 - 3n + 2} \right)^{\frac{n^2 + 1}{2n - 3}}$$

Determine the limit of the following sequence, which involves a difference of roots of different order (cube root and square root). To solve it, you will need to factorise a common term or use the technique of adding and subtracting an intermediate term to split the expression into two manageable limits:

$$\lim_{n \to \infty} \left( \sqrt[3]{n^3 + 3n^2} - \sqrt{n^2 - 4n} \right)$$

In the study of computer algorithms, it is essential to know the rate at which certain functions grow. Calculate the limit of the following sequence by comparing the orders of infinity of the different terms (factorials, exponentials and powers):

$$\lim_{n \to \infty} \frac{n! + 10^n}{n^n - 5^n}$$

Give a reasoned explanation of which term dominates in the numerator and which in the denominator to justify your result.

Use the Sandwich Theorem (or Squeeze Theorem) to find the value of the limit of the sequence $a_n$ defined by the following sum of fractions:

$$a_n = \frac{n}{n^2+1} + \frac{n}{n^2+2} + \frac{n}{n^2+3} + \dots + \frac{n}{n^2+n}$$

Find the values of the real parameters $a$ and $b$ such that the following equality between the limit of the sequence and the given value holds:

$$\lim_{n \to \infty} \left( \frac{n^2 + 5n + 1}{n - 2} - an - b \right) = 0$$

A sequence is defined recursively by the formula:

$$a_1 = \sqrt{6}$$ $$a_{n+1} = \sqrt{6 + a_n}$$

Assuming that the sequence is convergent (you can visualize its behaviour in the diagram below), calculate the exact value of its limit $L = \lim_{n \to \infty} a_n$.

A surveillance drone is programmed to make a series of jumps in a straight line. In the first second it travels $d$ metres. In each subsequent second, it travels a distance that is half the distance travelled in the previous second plus one extra metre to compensate for the wind. If $x_n$ is the total cumulative distance after $n$ seconds:

1. a) Find the expression for the general term of the distance travelled in each second $a_n$.
2. b) Determine the distance the drone will travel in a given second as $n$ tends to infinity (that is, $\lim_{n \to \infty} a_n$).

Given the sequence $a_n = \frac{3n + 2}{n + 1}$, prove using the formal definition of a limit ($\varepsilon$ and $N$) that its limit is $L = 3$.

To do so, for a given value of $\varepsilon = 0,001$, determine from which term $N$ of the sequence it holds that $|a_n - L| < \varepsilon$.

We want to find the limit of the sequence formed by the following sum of $n$ terms:

$$S_n = \sum_{k=1}^{n} \frac{1}{\sqrt{n^2 + k}} = \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2}} + \dots + \frac{1}{\sqrt{n^2 + n}}$$

To solve it, two 2nd year Bachillerato students propose different approaches:

• Javier's proposal: "Since the limit of a sum is the sum of the limits, I calculate the limit of each term as $n \to \infty$. Since $\lim_{n \to \infty} \frac{1}{\sqrt{n^2 + k}} = 0$ for any $k$, the sum of all limits gives $0 + 0 + \dots + 0 = 0$. Therefore, $\lim_{n \to \infty} S_n = 0$".

• Elena's proposal: "Javier's method is not valid because the number of terms is not fixed, but depends on $n$ and tends to infinity. We must bound the entire sum by replacing all denominators with the largest and the smallest to then apply the squeeze theorem".

1. a) Analyze Javier's proposal: explain why the algebraic property of the limit of a sum cannot be applied in this case and propose a simple counterexample where a sum of $n$ terms that individually tend to zero converges to a value different from zero.
2. b) Apply Elena's proposal: determine with reasoning two sequences $b_n$ and $c_n$ such that $b_n \le S_n \le c_n$, calculate their respective limits as $n \to \infty$, and deduce the true value of $\lim_{n \to \infty} S_n$.

Consider the sequence with roots of different index:

$$a_n = \sqrt{n^2 + 6n} - \sqrt[3]{n^3 + 3n^2}$$

1. a) Estimate: For very large values of $n$, approximate the behavior of each term by completing squares and cubes: note that $n^2 + 6n = (n+3)^2 - 9 \approx (n+3)^2$ and that $n^3 + 3n^2 = (n+1)^3 - (3n+1) \approx (n+1)^3$. From this informal asymptotic approximation, propose a reasoned estimation of the limit of the sequence before operating.
2. b) Verify: Rigorously prove the exact value of the limit by adding and subtracting the auxiliary term $n$ to transform the expression into the difference of two manageable blocks:

$$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left[ \left(\sqrt{n^2 + 6n} - n\right) - \left(\sqrt[3]{n^3 + 3n^2} - n\right) \right]$$

Multiply and divide each block by its corresponding conjugate expression (using the identity $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$ for the cubic block), calculate the limit of each part, and verify whether the result matches your estimation.

1. c) Justify the typical error: Explain with reasoning why a procedure that only keeps the highest-degree term inside each radical (doing $\sqrt{n^2} - \sqrt[3]{n^3} = n - n = 0$) leads to an incorrect conclusion, detailing why in a subtraction that generates an indeterminate form $\infty - \infty$ it is not permitted to ignore lower-degree terms.