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Question

# Consider the sequence $\left\{A_n\right\}$ whose nth term is given by$A_n=P\left(1+\frac{r}{12}\right)^n$where P is the principal, $A_n$ is the amount at compound interest after n months, and r is the interest rate compounded annually. Is $\left\{A_n\right\}$ a convergent sequence? Explain.

Solution

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$\color{#4257b2}\{ A_n \}is not a convergent sequence$ .

Since $r > 0$, we know that $1 + \dfrac{r}{12} > 1$ which means that

$\displaystyle \lim_{n \to \infty} \left( 1 + \dfrac{r}{12} \right)^n = \infty$

Therefore, $\{ A_n \}$ cannot converge.

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