TAOCP 1.2.11.3: Some Asymptotic Calculations
Section 1.2.11.3 exercises: 14/14 solved.
Section 1.2.11.3. Some Asymptotic Calculations
Exercises from TAOCP Volume 1 Section 1.2.11.3: 14/14 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [HM20] | hm-medium | verified | 1m21s |
| 2 | [HM20] | hm-medium | verified | 58s |
| 3 | [M20] | math-medium | verified | 1m10s |
| 4 | [HM10] | hm-simple | verified | 1m21s |
| 5 | [HM24] | hm-medium | verified | 3m57s |
| 6 | [HM20] | hm-medium | solved | 1m55s |
| 7 | [HM30] | hm-hard | solved | 6m22s |
| 8 | [HM30] | hm-hard | solved | 5m38s |
| 15 | [HM20] | hm-medium | solved | 2m08s |
| 16 | [M24] | math-medium | solved | 5m13s |
| 17 | [HM29] | hm-hard | verified | 1m56s |
| 18 | [M25] | math-medium | verified | 4m43s |
| 19 | [HM30] | hm-hard | verified | 6m56s |
| 20 | [HM30] | hm-hard | verified | 5m13s |
TAOCP 1.2.11.3 Exercise 1
Equation (5) states that f(x) = \sum_{k=0}^{n}\frac{f^{(k)}(0)}{k!
TAOCP 1.2.11.3 Exercise 2
We are asked to derive the series expansion \gamma(a,x) = \sum_{k\ge 0} \frac{(-1)^k x^{k+a}}{k!
TAOCP 1.2.11.3 Exercise 3
From Eq.
TAOCP 1.2.11.3 Exercise 4
**Corrected Solution to Exercise 1.
TAOCP 1.2.11.3 Exercise 5
We are asked to show that the remainder term $R$ in Eq.
TAOCP 1.2.11.3 Exercise 6
Write \frac{(n+\alpha)^n}{n^{\,n+\beta}} = n^{\alpha-\beta}\left(1+\frac{\alpha}{n}\right)^n.
TAOCP 1.2.11.3 Exercise 7
Let J(x)=\int_0^{y x^{1/4}} e^{-u}\left(1+\frac ux\right)^x\,du, where $y$ is fixed and $x\to\infty$.
TAOCP 1.2.11.3 Exercise 8
Let S_\alpha(n)=\sum_{k=0}^{n} k^{n+\alpha}e^{-k}, \qquad \alpha \ \text{fixed}.
TAOCP 1.2.11.3 Exercise 15
Let I_n=\int_0^\infty \left(1+\frac{z}{n}\right)^n e^{-z}\,dz .
TAOCP 1.2.11.3 Exercise 16
We are asked to prove the identity \sum_{k=0}^{n} (-1)^k \binom{n}{k} k^{\,n-1} Q(k) = (-1)^n (n-1)!
TAOCP 1.2.11.3 Exercise 17
Define S(n)=\sum_{k\ge0}\frac{(n+k-1)!
TAOCP 1.2.11.3 Exercise 18
Let Q(n)=\frac{1}{n!