Exercises: Sequences; limit of a sequence

Given the sequence whose general term is $a_n = \frac{1}{n^2}$, write the first four terms and 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" of a post in the first few hours follows the progression $c_n = \frac{4n^2}{n+2}$, where $n$ is the number of hours since publication. What happens to the amount 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 analysing the term of highest degree:

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

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

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

It is estimated that the value of a laptop 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 depreciation of the equipment.

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

Identify which of the following expressions corresponds to the definition of the number $e$ and calculate its limit:

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

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

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

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 the behavior of the sequence, observe the following diagram where the difference between the curve of the root and its linear asymptote is represented:

An investor analyses the growth of a cryptocurrency whose daily evolution follows the law of an exponential sequence. 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 relationship with the number $e$.

Find the value of the real parameters $k$ and $t$ so that the following equality is verified:

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

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

Study the behavior of the recursive sequence defined by:

$a_1 = 2$

$a_{n+1} = \frac{1}{2} \left( a_n + \frac{4}{a_n} \right)$

If we assume that the sequence is convergent and that its limit is $L$ ($L > 0$), calculate this value by solving the equation $L = \frac{1}{2} \left( L + \frac{4}{L} \right)$. 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 bacteria population as time tends to infinity. Reason what happens to the term $\text{sen}(n)$ in the calculation of the limit.

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

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

Indicate whether the sequence is divergent or convergent after solving the indeterminate form $\infty - \infty$.

Calculate the value of the following limit of a sequence that presents 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 indices (cube and square). To solve it, you must force the appearance of a common factor or use the technique of adding and subtracting an intermediate term that allows the expression to be separated into two manageable limits:

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

In the study of computing algorithms, it is essential to know the speed 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):

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

Explain with reasoning which term dominates in the numerator and which in the denominator to justify your result.

Use the Squeeze Theorem (or Sandwich 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 $b$ so that the following equality between the limit of the sequence and the indicated value is satisfied:

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

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 $d$ feet. In each subsequent second, it travels a distance that is half the distance traveled in the previous second plus an extra foot to compensate for the wind. If $x_n$ is the total accumulated 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 when $n$ 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 this, for a value of $\varepsilon = 0.001$, determine from which term $N$ of the sequence it is satisfied that $|a_n - L| < \varepsilon$.