True or False: The sequence $a_n=1/n$ converges faster than the sequence $b_n=1/n^2$.

Explain.

Answer. False. Both converge linearly (order $p=1$) with rate $r=1$. Let $x_n=1/n^k$

Choose most accurate floating point expression if a,b ~ eps (machine epsilon)

(a) $x=1+(a+b)$

(b) $x=(1+a)+b$

(c) $x=(1+b)+a$

Explain.

Answer. (a) since $(a+b)$ might have carry over digits, whereas for $a=b=ϵ$, $(1+a)+b=(1+b)+a=0$ in $𝔽$.

True or False: The sequence $a_n=1/n^2$ converges quadratically. Explain.

Answer. False, see (1).

True or False: It is possible to verify Cauchy sequence convergence computationally. Answer(s). False. Proof of convergence of ${\left\{a_n\right\}}_{n\in \mathbb{N}}$ requires consideration of $n\to \infty$. True. Verification is done for finite $n$.

In floating point arithmetic, the result of 1/0 - 1/0 is:
(a) Infinity
(b) Zero
(c) NaN
Explain.

Answer. (c), Infinity-Infinity is not defined, hence NaN