### Exercise

#### 4.9

If $\alpha<\beta,$ show that $\alpha^{N}$ is exponentially small relative to $\beta^{N}$. For $\beta=1.2$ and $\alpha=1.1,$ find the absolute and relative errors when $\alpha^{N}+\beta^{N}$ is approximated by $\beta^{N},$ for $N=10$ and $N=100 .$

#### 4.71

Show that $P(N)=\sum_{k \ge 0} \frac{(N-k)^{k}(N-k) !}{N !}=\sqrt{\pi N / 2}+O(1)$

### Problem

#### 1

Which of the following is an asymptotic expansion of $e^{H_{N}} ?$

• $1-\frac{1}{2 N}+O\left(\frac{1}{N^{2}}\right)$
• $N+O(1)$
• $1+\frac{1}{N}+O\left(\frac{1}{N^{2}}\right)$
• $1+\frac{1}{2 N}+O\left( \frac{1}{N^{2}}\right)$
• $1-\frac{1}{N}+O\left(\frac{1}{N^{2}}\right)$

#### 2

Which of the following has approximate value $1.22019 ?$

• $1.01^{10}$
• $1.05^{10}$
• $1.01^{20}$
• $1.01^{50}$
• $1.01^{100}$