Hi MHF.
I would like to see some proof/detailed explanation for
Where H is the starting amount, i is the percentage and m is the # of times per year where growth is calculated and n is the number of years. e is Euler's number.
The formula you want, in terms of the variables you have defined, is:
To prove this statement requires some methods usually found in a second semester of elementary calculus. To begin, let's write:
Because the constant does not depend on , we may divide through by this to obtain:
Now, taking the natural log of both sides, we have:
The log of a limit is equal to the limit of a log, hence:
Applying the property of logs we have:
Dividing through by :
Bring to the denominator:
We now have the indeterminate form 0/0, thus application of L'Hôpital's rule reveals:
Simplify:
We see now the term involving vanishes to zero, hence:
Solve for