TAOCP 1.2.11.2: Euler's Summation Formula
Section 1.2.11.2 exercises: 13/13 solved.
Section 1.2.11.2. Euler's Summation Formula
Exercises from TAOCP Volume 1 Section 1.2.11.2: 13/13 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [M18] | math-medium | verified | 1m44s |
| 2 | [HM20] | hm-medium | verified | 2m23s |
| 3 | [HM20] | hm-medium | verified | 1m50s |
| 4 | [HM20] | hm-medium | verified | 43s |
| 5 | [HM30] | hm-hard | verified | 59s |
| 6 | [HM30] | hm-hard | verified | 4m03s |
| 7 | [HM32] | hm-hard | verified | 44s |
| 8 | [M23] | math-medium | verified | 45s |
| 9 | [M25] | math-medium | solved | 3m08s |
| 10 | [HM22] | hm-medium | verified | 45s |
| 11 | [M11] | math-simple | verified | 44s |
| 12 | [HM25] | hm-medium | solved | 3m10s |
| 13 | [M10] | math-simple | verified | 1m31s |
TAOCP 1.2.11.2 Exercise 1
We are asked to **prove Eq.
TAOCP 1.2.11.2 Exercise 2
In Section 1.
TAOCP 1.2.11.2 Exercise 3
Let $m=2k>0$.
TAOCP 1.2.11.2 Exercise 4
Let $f(x) = x^m$, where $m$ is a nonnegative integer.
TAOCP 1.2.11.2 Exercise 5
Assume n!
TAOCP 1.2.11.2 Exercise 6
Let S(x)=\sqrt{2\pi x}\left(\frac{x}{e}\right)^x .
TAOCP 1.2.11.2 Exercise 7
Let P_n = 1^1 2^2 3^3 \cdots n^n = \prod_{k=1}^n k^k.
TAOCP 1.2.11.2 Exercise 8
Let m = an^2 + bn, \qquad a>0.
TAOCP 1.2.11.2 Exercise 9
The error is not in the mechanics of exponentiating $e^x$, but in the claim that the logarithmic expansion was complete at order $n^{-3}$.
TAOCP 1.2.11.2 Exercise 10
Let $u(z)=O(z^m)$ as $z\to 0$, with $m>0$.
TAOCP 1.2.11.2 Exercise 11
Equation (18) states \sqrt[n]{n} = e^{\ln n/n} = 1 + \frac{\ln n}{n} + O\!
TAOCP 1.2.11.2 Exercise 12
The critical issue is not the manipulation of generating functions but the implicit assumption that the coefficient sequence is governed by a finite radius of convergence.