Algebra

Exponential Theorem

𝑎𝑥=1+𝑐𝑥+𝑐2𝑥22!+𝑐3𝑥33!+⋯149 where 𝑐=(𝑎−1)−12(𝑎−1)2+13(𝑎−1)3−⋯ Proof: 𝑎^𝑥={1+(𝑎−1)}^𝑥. Expand this by Binomial Theorem, and collect the coefficients of 𝑥; thus 𝑐 is obtained. Assume 𝑐₂, 𝑐₃, ⋯ as the coefficients of the succedding powers of 𝑥, and with this assumption write out the expansions of 𝑎_𝑥, 𝑎_𝑦 and 𝑎_𝑥+𝑦. Frm the product of the first two series, which product must be equivalent to the third. Therefore equate the coefficient of 𝑥 in this product with that in the expansion of 𝑎_𝑥+𝑦. In the identity so obtained, equate the coefficients of the successive powers of 𝑦 to determine 𝑐₂, 𝑐₃, ⋯.

Exponential Function

Let 𝑒 be that value of 𝑎 which makes 𝑐=1, then 𝑒𝑥=1+𝑥+𝑥22!+𝑥33!+⋯150

Exponential

𝑒=1+1+122!+133!+⋯151  =2.718281828⋯ Proof: By making 𝑥=1 in (150)

Logarithm

By makng 𝑥=1 in (149) and 𝑥=𝑐 in (150), obtain 𝑎=𝑒^𝑐; that is, 𝑐=log_𝑒𝑎152 Therefore by (149) log_𝑒𝑎=(𝑎−1)−12(𝑎−1)²+13(𝑎−1)³−⋯154 By substitution, log(1+𝑥)=𝑥−𝑥²2+𝑥³3−𝑥⁴4+⋯155 log(1−𝑥)=−𝑥−𝑥²2−𝑥³3−𝑥⁴4−⋯156 ∴ log1+𝑥1−𝑥=2𝑥+𝑥³3+𝑥⁵5+⋯157 Put 𝑚−1𝑚+1 for 𝑥 in (157); thus, log 𝑚=2𝑚−1𝑚+1+13𝑚−1𝑚+1³+15𝑚−1𝑚+1⁵+⋯158 Put 12𝑛+1 for 𝑥 in (157); thus, log(𝑛+1)−log 𝑛=212𝑛+1+13(2𝑛+1)³+15(2𝑛+1)⁵+⋯159

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive