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 $v_n = 1200 \times (0,8)^n$. Calculate the limit of this sequence and interpret what the result means in the context of the device's depreciation.

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

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.

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

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

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 $\infty - \infty$.

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 behavior 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 $d$ meters. In each subsequent second, it travels a distance that is half the distance traveled in the previous second plus one extra meter 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 traveled 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 $|a_n - L| < \varepsilon$.