Mathematics & Numerical Analysis

A fundamental theorem of real analysis states: 'Every convergent sequence of real numbers is bounded.' Thus, boundedness is a necessary condition for convergence (though not sufficient, e.g., $(-1)^n$ is bounded but not convergent).

Detailed analysis of options:

• Bounded is the correct necessary condition; an unbounded sequence can never converge.
• Having more than one limit is impossible because the limit of a convergent sequence is unique.
• Monotonic is not necessary; for example, $a_n = \frac{(-1)^n}{n}$ converges to 0 but oscillates and is not monotonic.
• Not Cauchy is incorrect because every convergent sequence in $\mathbb{R}$ must be a Cauchy sequence.

Behavior of real sequences:

• $(a_n) = (-1)^n$ does not converge because it oscillates finitely between $-1$ and $+1$.
• $(b_n) = 2^n$ does not converge because it grows without bound to $+\infty$.
• Neither sequence converges to a finite real limit.
• $(a_n) = (-1)^n$ is an oscillatory sequence with two distinct subsequential limits ($-1$ and $+1$), hence divergent. $(b_n) = 2^n$ tends to $+\infty$, hence divergent. Therefore, both $(a_n)$ and $(b_n)$ are divergent.

Convergence criteria for geometric sequences:

• For $r = 2$, $2^n \to \infty$, which diverges.
• For $r = 0.5$, $(0.5)^n \to 0$, which converges.
• For any real number $r$ satisfying $|r| \lt 1$ (i.e. $-1 \lt r \lt 1$), $\lim_{n \to \infty} r^n = 0$. Hence the sequence is strictly convergent for any $|r| \lt 1$.
• The sequence converges to $0$ when $|r| \lt 1$, not diverges.

Evaluation of limit of algebraic sequence:

• $1$ neglects the factor of $2$ in the AP sum formula.
• The sum of the first $n$ natural numbers is given by $\sum_{k=1}^n k = \frac{n(n + 1)}{2}$. Substituting into the expression: $\lim_{n \to \infty} \frac{n(n + 1)}{2 n^2} = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{2} = \frac{1 + 0}{2} = \frac{1}{2}$.
• $0$ would occur if the denominator had higher degree.
• $2$ is the reciprocal.