The (almost really) Complete Works of Lewis Carroll

Formulæ

Source: cyclostyled 1878

e, as series	$1 + 1 + \frac{1}{2} + \frac{1}{3} + \dots$
e, in decimals	$2⋅718281828 \dots$
$e^{x}$ , as series	$1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + \dots$
${log}_{e} a$ , do.	$(a - 1) - \frac{{(a - 1)}^{2}}{2} + \frac{{(a - 1)}^{3}}{3} - \dots$
$a^{x}$ , do.	$1 + {log}_{e} a x + \frac{{({log}_{e} a)}^{2} . x^{2}}{2} + \dots$
${log}_{e} (a + 1)$ , do.	$a - \frac{a^{2}}{2} + \frac{a^{3}}{3} - \dots$
${log}_{e} (a + 1) - {log}_{e} a$	$2 {\frac{1}{2 a + 1} + \frac{1}{3 . {(2 a + 1)}^{3}} + \frac{1}{5 . {(2 a + 1)}^{5}} + \dots$
${log}_{e} 10$ , in decimals	2⋅3025851
${log}_{10} e$ ,do.	0⋅4342945

$cos θ$ , in terms of `θ`	$1 - \frac{θ^{2}}{2} + \frac{θ^{4}}{4} - \dots$
$sin θ$ ,do.	$θ - \frac{θ^{3}}{3} + \frac{θ^{5}}{5} - \dots$
$cos θ$ , exponential value	$\frac{e^{θ i} + e^{- θ i}}{2}$ , [ $i = \sqrt{- 1}$ ]
$sin θ$ ,do.	$\frac{e^{θ i} - e^{- θ i}}{2 i}$
${tan}^{- 1} x$ , in terms of `x`	$x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \dots$

`π`, approximate values	$\frac{22}{7}$ , $\frac{355}{113}$
`π`, in decimals	3⋅1415927
$\frac{π}{180}$ ,do.	0⋅0174533
$\frac{180}{π}$ ,do.	57⋅2957795

Mar. 19, 1878.