Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/391

379

DES SUITES.

APPLICATION II.

16. On sait qu’en faisant usage des logarithmes naturels on a

$\operatorname {Log} .(1+a)={\frac {a}{1}}-{\frac {a^{2}}{2}}+{\frac {a^{3}}{3}}-{\frac {a^{4}}{4}}+{\frac {a^{5}}{5}}-\ldots$

de sorte qu’ici

$\operatorname {f} [a]=\operatorname {Log} .(1+a)$

donc, 1.^o la somme de la série

${\frac {a\operatorname {Cos} .t}{1}}-{\frac {a^{2}\operatorname {Cos} .2t}{2}}+{\frac {a^{3}\operatorname {Cos} .3t}{3}}-{\frac {a^{4}\operatorname {Cos} .4t}{4}}+\ldots$

sera (Remarque I)

${\frac {\operatorname {Log} .\left\{1+a\left(\operatorname {Cos} .t+{\sqrt {-1}}\operatorname {Sin} .t\right)\right\}+\operatorname {Log} .\left\{1+a\left(\operatorname {Cos} .t-{\sqrt {-1}}\operatorname {Sin} .t\right)\right\}}{2}},$

ou bien

${\frac {\operatorname {Log} .\left\{(1+a\operatorname {Cos} .t)+{\sqrt {-1}}a\operatorname {Sin} .t\right\}\left\{(1+a\operatorname {Cos} .t)-{\sqrt {-1}}a\operatorname {Sin} .t\right\}}{2}}$

$={\frac {\operatorname {Log} .\left\{(1+a\operatorname {Cos} .t)^{2}+a^{2}\operatorname {Sin} .^{2}t\right\}}{2}}$

ou enfin

${\frac {\operatorname {Log} .\left(1+2a\operatorname {Cos} .t+a^{2}\right)}{2}}=\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .t+a^{2}}}\,;$

2.^o la somme de la série

${\frac {a\operatorname {Sin} .t}{1}}-{\frac {a^{2}\operatorname {Sin} .2t}{2}}+{\frac {a^{3}\operatorname {Sin} .3t}{3}}-{\frac {a^{4}\operatorname {Sin} .4t}{4}}+\ldots$

sera (Remarque II)

${\frac {\operatorname {Log} .\left\{1+a\left(\operatorname {Cos} .t+{\sqrt {-1}}\operatorname {Sin} .t\right)\right\}-\operatorname {Log} .\left\{1+a\left(\operatorname {Cos} .t-{\sqrt {-1}}\operatorname {Sin} .t\right)\right\}}{2{\sqrt {-1}}}},$

ou bien