Real Analysis: Sequences and Convergence

Using Cauchy's criterion of convergence, examine the convergence of sequence $f_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!}$.

Using Cauchy's criterion of convergence, examine the convergence of sequence $f_n = \frac{1}{1!} - \frac{1}{2!} + \frac{1}{3!} - \ldots + (-1)^{n+1} \frac{1}{n!}$, also find the limit.

Given a sequence $(a_n)$ and its subsequence $(a_{n_k})$, prove that the kth term of the subsequence is at least k terms along in the original sequence, i.e., $n_k \geq k$.

Prove that if a sequence $(a_n)$ diverges to infinity, then all of its subsequences also diverge to infinity.

Consider the recursively defined sequence $a_{n+1} = \frac{1}{3} a_n + 1$ with $a_1 = 1$. Use the Monotone Convergence Theorem to show that this sequence converges.

Given the recursive sequence defined by $a_1 = 1$ and $a_n = 1 + \frac{1}{1 + a_{n-1}}$ for $n \geq 2$, determine the limit of the sequence as $n \to \infty$.

Prove that the sequence $\frac{3n+1}{n+2}$ converges to the limit 3 using the epsilon definition of the limit of a sequence.

Make the quantity $\frac{3n + 2}{2n + 5} - \frac{3}{2}$ less than epsilon for $n > N$ in an epsilon-N proof.

Find the Taylor series centered at a given value $a$ and find the associated radius of convergence for $f(x) = e^{2x}$ at $a = 6$.