4.E: Convergencia de Secuencias y Series (Ejercicios)
- Page ID
- 109474
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Demuéstralo si\(\lim_{n \to \infty }s_n = s\) entonces\(\lim_{n \to \infty }\left |s_n \right | = \left |s \right |\). Demostrar que lo contrario es cierto cuando\(s = 0\), pero no necesariamente es cierto de lo contrario.
Q2
- Dejar\((s_n)\) y\((t_n)\) ser secuencias con\(s_n ≤ t_n,∀n\). Supongamos\(\lim_{n \to \infty }s_n = s\) y\(\lim_{n \to \infty }t_n = t\). Demostrar\(s ≤ t\). [Pista: Asumir por contradicción, eso\(s > t\) y usar la definición de convergencia con\(ε = \(frac{s-t}{2}\) para producir un\(n\) con\(s_n > t_n\).]
- Demostrar que si una secuencia converge, entonces su límite es único. Es decir, probarlo si\(\lim_{n \to \infty }s_n = s\) y\(\lim_{n \to \infty }s_n = s\), entonces\(s = t\).
Q3
Demostrar que si la secuencia\((s_n)\) está acotada entonces\(\lim_{n \to \infty }\left (\frac{s_n}{n} \right ) = 0\).
Q4
- Demostrar que si\(x \neq 1\), entonces\[1 + x + x^2 +\cdots + x^n = \frac{1 - x^{n+1}}{1-x}\]
- Utilice (a) para probar que si\(|x| < 1\), entonces\(\lim_{n \to \infty }\left ( \sum_{j=0}^{n} x^j \right ) = \frac{1}{1-x}\)
Q5
Demostrar\[\lim_{n \to \infty }\frac{a_0 + a_1n + a_2n^2 +\cdots + a_kn^k}{b_0 + b_1n + b_2n^2 +\cdots + b_kn^k} = \frac{a_k}{b_k}\]
siempre\(b_k \neq 0\). [Observe que como un polinomio solo tiene finitamente muchas raíces, entonces el denominador será distinto de cero cuando n sea suficientemente grande.]
Q6
Demostrar que si\(\lim_{n \to \infty }s_n = s\) y\(\lim_{n \to \infty }(s_n - t_n) = 0\), entonces\(\lim_{n \to \infty }t_n = s\).
Q7
- Demostrar que si\(\lim_{n \to \infty }s_n = s\) y\(s < t\), entonces existe un número real\(N\) tal que si\(n > N\) entonces\(s_n < t\).
- Demostrar que si\(\lim_{n \to \infty }s_n = s\) y\(r < s\), entonces existe un número real\(M\) tal que si\(n > M\) entonces\(r < s_n\).
Q8
Supongamos que\((s_n)\) es una secuencia de números positivos tal que\(\lim_{n \to \infty }\left ( \frac{s_{n+1}}{s_n} \right ) = L\)
- Demostrar que si\(L < 1\), entonces\(\lim_{n \to \infty }s_n = 0\). [Pista: Elija\(R\) con\(L < R < 1\). Por el problema anterior,\(∃\; N\) tal que si\(n > N\), entonces\(\frac{s_{n+1}}{s_n} < R\). \(n_0 > N\)Déjese arreglar y mostrar\(s_{n_0+k} < R^ks_{n_0}\). Concluya eso\(\lim_{k \to \infty }s_{n_0+k} = 0\) y deje\(n = n_0 + k\).]
- \(c\)Sea un número real positivo. Demostrar\[\lim_{n \to \infty }\left ( \frac{c^n}{n!} \right ) = 0\]