Exercises: Limit of a sequence

Determine whether the sequence whose general term is $a_n = \frac{(-1)^n \cdot 5}{n}$ 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 the denominator:

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

Determine the value towards which the sequence $e_n$ tends for very large values of $n$:

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

Study the behavior and calculate the limit of the exponential sequence:

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

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:

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

The temperature in degrees Fahrenheit of a piece of metal cooling in a room follows the sequence $T_n = 22 + 85 \cdot (0.82)^n$, 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:

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

A population of bacteria cultured in a laboratory grows according to the mathematical model $P_n = \frac{150000n}{3n + 250}$, where $n$ represents the number of days since the start of the culture. Determine the carrying capacity of the environment, that is, the limit of the bacteria population in the long term.

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

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

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

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

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

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. Knowing 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 constructed in this way: $S_\infty = \lim_{n \to \infty} (A_1 + A_2 + \dots + A_n)$.

Find the values of the real parameters $a$ and $b$ (with $b > 0$) so that the following equality is satisfied:

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

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 out from which term $n$ it is satisfied that the distance between the terms of the sequence and its limit is strictly less than one thousandth, that is, $|a_n - 4| < 0.001$.

Resolve the following limit that gives rise 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 $L_0 = 1$ foot (panel A), at each step $n$ the total length of the curve is multiplied by $\frac{4}{3}$ with respect 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 presents a double square root and requires applying the conjugate technique carefully:

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

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 $C_n = 15 + C_{n-1} \cdot 0.4$, where $15 \text{ mg/L}$ is the contribution of the new dose and the factor $0.4$ indicates that a $40\%$ of the drug from the previous day remains. If the initial concentration is $C_1 = 15$, calculate the equilibrium concentration in the blood in the long term, that is, $\lim_{n \to \infty} C_n$.

A sequence is defined by the following recurrence law: 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 (so it is convergent and has a limit $L$), set up the equation that the limit must satisfy and find it.

A high-tech elastic cable is subjected to iterative tension tests. Its initial length at rest is $20$ feet. In the first test it stretches $4$ feet. In each successive test, the material loses elasticity and stretches a $75\%$ of what it stretched in the immediately preceding test. The total length after $n$ tests is given by $L_n = 20 + \sum_{k=1}^n 4 \cdot (0.75)^{k-1}$. Calculate the theoretical maximum length that the cable will reach if subjected to infinite tests.