proposition 7.52 Alternating series test
open in the book ·
parts/02-mathematical-methods/05-real-analysis.tex:1104
· p. 220
Rests on
-
depends_on
corollary A.46
Monotone convergence
¶
-
depends_on
theorem A.40
Least-upper-bound property
¶
- depends_on definition A.36 Cut ¶
- depends_on definition A.38 Order on $\R$ ¶
- proves proof app:A-long-proofs@proof-29 ¶
- proves proof app:A-long-proofs@proof-33 ¶
-
depends_on
theorem A.40
Least-upper-bound property
¶
-
depends_on
definition 7.45
Series
¶
- depends_on definition 7.2 Absolute value ¶
-
depends_on
definition 7.4
Convergence
¶
- depends_on definition 7.2 Absolute value ¶ ↺
- proves proof ch:05-real-analysis@proof-33 ¶
Supports
-
depends_on
lemma 7.73
$\cos 2 < 0$
¶
-
depends_on
definition 7.74
$\pi$
¶
-
depends_on
lemma 7.75
The first quadrant
¶
- depends_on proposition 7.76 $\pi$ as the circle constant ¶
-
depends_on
proposition 7.77
Special values, periodicity, and the kernel
¶
- depends_on lemma 7.78 Chord, arc, tangent ¶
- depends_on proposition 7.85 Irrationality of $\pi$ ¶
- depends_on proposition 7.79 The polygon recursion ¶
- depends_on proposition 7.83 Viète's product ¶
- depends_on proposition 125.2 Crystallographic restriction ¶
- depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
- depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
- depends_on proposition 7.79 The polygon recursion ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
- depends_on proposition 125.2 Crystallographic restriction ¶ ↺
-
depends_on
lemma 7.75
The first quadrant
¶
-
depends_on
definition 7.74
$\pi$
¶
Neighborhood
Every logical edge within two steps of this node.
- declared and complete
- partly declared
- a check failed
- not graded
- declared in the source
- inferred from structure
Edges
| type | direction | node | provenance | where |
|---|---|---|---|---|
depends_on |
→ | Monotone convergence | declared | parts/02-mathematical-methods/05-real-analysis.tex:1109 |
depends_on |
→ | Series | declared | parts/02-mathematical-methods/05-real-analysis.tex:1109 |
depends_on |
← | $\cos 2 < 0$ | declared | parts/02-mathematical-methods/05-real-analysis.tex:1907 |
proves |
← | ch:05-real-analysis@proof-33 | declared | parts/02-mathematical-methods/05-real-analysis.tex:1112 |