I am having trouble deciphering a portion from my finance text.
Let $i = \text{interest rate}$, $n = \text{Some arbitrary time period}$ and $C = \text{Cash invested}$
And also
$C(1+i)^n$ (compound interest) $\dots (1)$
$C(1+in)$ (simple interest) $\dots (2)$
It is comparing compound and simple interest, the section reads as follows:
"It is interesting to note the connection between the Taylor expansion of the formula for an n-year accumulation of a unit investment under compound interest, $(1)$, and that for an accumulation under simple interest, $(2)$.
$C(1+i)^n = C(1 + in + \mathcal{O}(i^2))$
In particular, we see that, for small compound interest rates, the higher order terms are negligible and the two expressions are approximately equal."
I am not sure how they obtained the taylor expansion of $C(1+i)^n$ and what $\mathcal{O}(i^2)$ really even represents.
I know that the taylor expansion for compound interest looks very similar to the formula for simple interest, bar the $\mathcal{O}$ part, but what does it mean "Higher order terms are negligible?".
This is a Binomial_series $$(1+i)^n= 1 + ni +\frac{n(n-1)}{2!}i^2 + i^3(...)$$ If $i$ is 'small' (e.g. $ni < 0.1$) you can 'omit' higher powers of $i$. Example with two decimals $n=10,\, i=1\%=0.01,\,$ and for simplicity $C=1:$ $$(1+0.01)^{10} = 1.104622\dots \approx 1.10$$ $$1 + 10\times 0.01 +\frac{10\times 9}{2}0.01^2 + \dots =1 + 0.1+0.0045 +\dots \approx 1.10$$
For a bit more information about $O(i^2)$ see the examples in Taylor_series and their reference to the Big_O_notation.