7 - ...
This result may be proved in various ways: it may also be verified by calculation. To do this, remember that if
|
a1 b1+ | a2 b2+ | a3 b3+ ... | an bn | = | Pn Qn | ; then |
| P1=a1, | P2=b2 P1, | P3=b3 P2+a3 P1, | P4=b4 P3+a4 P2, etc. |
| Q1=b1, | Q2=b2 Q1+a2, | Q3=b3 Q2+a3 Q1, | Q4=b4 Q3+a4 Q2, etc. |
in the case before us we have
| a1=x, | a2=-x2, | a3=-x2, | a4=-x2, | a5=-x2, etc. |
| b1=1, | b2=3, | b3=5, | b4=7, | b5=9, etc. |
| P1=x | Q1=1 |
| P2=3x | Q2=3-x2 |
| P3=15x-x3 | Q3=15-6x2 |
| P4=105x-10x3 | Q4=105-45x2+x4 |
| P5=945x-105x3+x5 | Q5=945-420x2+15x4 |
| P6=10395x-1260x3+21x5 | Q6=10395-4725x2+210x4-x6 |
We can use this algebraically, or arithmetically. If we divide Pn by Qn, we shall find a series agreeing with the known series for tan x, as far as n terms. That series is
| x + | x3 3 | + | 2x5 15 | + | 17x7 315 | + | 62x9 2835 | + ... |