Show that, as the time interval of compounding goes to zero,...

Show that, as the time interval of compounding goes to zero, the bank balance after $k$ years approaches an exponential function $e^{r k} a$, where $r$ is the yearly interest rate and $a$ is the initial balance.

Key Concepts

Limit Definition of the Exponential Function

The limit definition of the exponential function is a fundamental mathematical idea that expresses the exponential function as the limit of (1 + x/n)^n as n approaches infinity. This concept is pivotal in bridging the gap between discrete compounding and continuous compounding, thereby explaining why the continuously compounded growth converges to an exponential model.

Exponential Growth

Exponential growth describes processes where the rate of change of a quantity is directly proportional to the current amount. In the context of finance, this is exemplified by continuously compounded interest, where the accumulation of interest results in growth that can be modeled by an exponential function.

Compound Interest

Compound interest refers to the process where interest is calculated on both the initial principal and the accumulated interest from previous periods. This principle is foundational in understanding how investments grow over time and provides the base for more advanced concepts like continuous compounding.

Continuous Compounding

Continuous compounding is an idealization in which interest is compounded an infinite number of times per period, effectively treating the compounding process as occurring continuously. This concept leads to the formulation of growth using the exponential function, representing a more precise model of how money accumulates over time under constant interest rates.

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