How to calculate compound interest with shrinking interest rates? The question arised when I was thinking about population growth. If a population grows by 1% we can simply plug into to the formula:
$$ \text{final population}= P \cdot (1+p)^n $$
where $P$ is the initial population, $p=0.01$ in the example and $n$ is the number of years.
Is there a similar (explicit) formula if $p$ shrinks at a constant rate, let's say for example after $n$ years $p_n = 0.99  p_{n-1}$ and $p_0 = 0.01$?
Of course, the formula $P \cdot \prod_{i=0}^{n}(1+p_i)$ would describe this, if we would also define the sequence $p_n$ but I want something that only takes $p$ as an input.
 A: Suppose $1$ dollar is deposited, and the interest rate decreases over time, such that after $n$ years the interest rate is $s^{n-1}r$ where $0<s<1.$
Assuming interest is compounded annually, the amount after $N$ years is:
$$A_N=\prod_{n=1}^N{(1+s^{n-1}r)}$$
This can be expressed in terms of the q-Pochhammer Symbol, but that is not helpful if we want to calculate or approximate its value with a simple calculator.
Here is how we can approximate it with a simple calculator:
$$A_N=\prod_{n=0}^{N-1}{(1+s^{n}r)}$$
$$=\exp{\sum_{n=0}^{N-1}\ln{{(1+s^{n}r)}}}$$
Use the Maclaurin series for $\ln{(1+s^{n}r)}$:
$$=\exp{\sum_{n=0}^{N-1}\sum_{k=1}^\infty{\frac{(-1)^{k+1}(s^nr)^k}{k}}}$$
Switch the order of summation:
$$=\exp{\sum_{k=1}^\infty\frac{(-1)^{k+1}r^k}{k}}\sum_{n=0}^{N-1}{(s^k)^n}$$
The second series is a geometric series:
$$=\exp{\sum_{k=1}^\infty\frac{(-1)^{k+1}r^k}{k}}\left(\frac{1-s^{kN}}{1-s^k}\right)$$
This series converges by the alternating series test. $r$ is an interest rate, so $r$ is usually close to $0$, so the series can be approximated by the sum of the first few terms. In fact, using only the first term gives a decent approximation (unless $r$ is large and/or $s$ is very close to $1$).
$$A_N\approx e^{r\left(\frac{1-s^N}{1-s}\right)}$$
Since $e^r\approx 1+r$:
$$A_N\approx (1+r)^{\frac{1-s^N}{1-s}}$$
Notice that $A_N$ converges as $N\to\infty$:
$$\lim_{N\to\infty}A_N \approx (1+r)^{\frac{1}{1-s}}$$
