Comment by necovek
16 hours ago
> Whether the infinite-series summation limits are worth knowing is questionable, though. That's more of historical interest.
A very common example is compound, continual interest accumulation. As a mathematician, I had trouble understanding how banks calculate it (as they would obviously not point me at the derivation of the formula), but it's actually a limit sum over infinitely small interest terms.
Similarly, these techniques work to explain and derive numerical methods for calculating many natural constants like e, pi etc.
The only intuitive definition of e I can remember is "how much $1 becomes after a year in a savings account with 100% interest compounded infinitely" funny enough.
I just call it “pi but for curves”
That definition also gives a nice introduction to e's taylor series!
> As a mathematician, I had trouble understanding how banks calculate it (as they would obviously not point me at the derivation of the formula)
This is not true? The formula is well known:
A = P*e^(rt)
where"
A = Final amount
P = Principal (initial investment/loan)
r = Annual interest rate (as a decimal)
t = Time in years
e = Euler's number
A loan is subdivided and sold in different allotments each year at a different interests rate, principal, and terms. You want to sell the entire debt at some point in the future. How do you calculate the price such that you don’t sell at a loss?
Why does that formula work? ;)
I did not say "derivation of the formula" by accident: this is about why would this produce something that makes sense as "interest-as-percentage"?
That is the integral
You're assuming banks use that formula and not just YOLO it, "haha fuck you pay me more".
If you learnt calculus, you could check their work and see that it's accurate. And it only takes algebra and a few minutes of work to see that your bank statements match the formula that's been given.
Where I live, the banks say that they compound interest daily, but pay interest monthly. So, I think the small interest term is just limited to a day where the error is small enough for the average person to not care.