Sequences; the limit of a sequence

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 $|a_n - L| < \varepsilon$.

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.

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

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

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

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:

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

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 $L = \lim_{n \to \infty} a_n$.

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?

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, $\lim_{n \to \infty} a_n$).

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:

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

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:

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