Series and the Geometric Series: Partial Sums and Convergence

A bond pays a coupon $C$ at the end of each period for $N$ periods and a face value $F$ at maturity. The price is $P = \sum_{k=1}^N C/(1+y)^k + F/(1+y)^N$ — a finite geometric series in the discount factor $1/(1+y)$. Taking $N \to \infty$ gives a perpetuity $P = C/y$, a result that falls out of the infinite geometric series formula $\sum_{n=0}^\infty ar^n = a/(1-r)$ for $|r| < 1$. This lesson builds the machinery: convergence via partial sums, the geometric series as the single most important series in finance, and the Divergence Test as the cheapest way to prove a series diverges.

Learning objectives. By the end of this note you should be able to: