search

Motorola M500 Manuel d'utilisateur Page 13

  • Télécharger
  • Ajouter à mon manuel
  • Imprimer
  • Page
    / 32
  • Table des matières
  • MARQUE LIVRES
    • Noté. / 5. Basé sur avis des utilisateurs
Vue de la page 12
M500 180 Page 11
Solution 178.7 – Series
Show that
x
1 − x
−
x
3
1 − x
3
+
x
5
1 − x
5
−. . . =
x
1 + x
2
+
x
2
1 + x
4
+
x
3
1 + x
6
+. . . .
John Bull
This cannot be true for all x. A simple inspection shows it to be undefined
for x = 1. So we proceed on the assumption that −1 < x < 1.
Call the LHS series S, and expand each term by the binomial theorem,
which is permissible if and only if −1 < x < 1. We then have
S = x + x
2
+ x
3
+ x
4
+ . . .
− x
3
− x
6
− x
9
− x
12
− . . .
+ x
5
+ x
10
+ x
15
+ x
20
− . . .
− x
7
− x
14
− x
21
− x
28
− . . .
+ . . . ,
where each term is expressed as a separate row. Now rearrange this so that
each column is expressed as a row, then
S = x − x
3
+ x
5
− x
7
+ . . .
+ x
2
− x
6
+ x
10
− x
14
+ . . .
+ x
3
− x
9
+ x
15
− x
21
+ . . .
+ x
4
− x
12
+ x
20
− x
28
+ . . .
+ . . . .
Each row can now be seen to be a binomial series expansion of each succes-
sive term on the RHS. Thus it is shown that LHS = RHS, as required.
Solved in a similar manner by Peter Fletcher.
Problem 180.2 – Unlimited prize
Paul Richards
A casino owner devises a gambling game for which he can advertise an
unlimited prize, but which also has a predictable distribution of payouts.
A gambler pays a stake, and the croupier throws a six-sided die repeat-
edly until all the numbers from 1 to 6 inclusive have appeared, irrespective
of repetition. The gambler receives winnings proportional to the number of
throws of the dice.
What is the correct stake to make the game fair?
Vue de la page 12
1 2 ... 8 9 10 11 12 13 14 15 16 17 18 ... 31 32

Commentaires sur ces manuels

Pas de commentaire
  • Central Machinery Hobo International Record Power Calorex Xaircraft
  • NEC Pièces de boîtier d'ordinateur ONKYO Routeurs HP Kits de voiture EdgeStar Accessoires de refroidissement du matériel Hach Non
  • LG VS415PP Weller WSD80 Sony ICD-PX312 Philips PET100
  • Olympus Stylus Zoom Instructions d'exploitation Velodyne VX-10 Manuel de l'utilisateur Nokia E61i Manuel de l'utilisateur Philips DVP3350/58 Manuel de l'utilisateur Summit FS20L Manuel de l'utilisateur