When does the geometric-series sum $a/(1-r)$ converge as $n$ approaches infinity?

The convergence condition is $|r| < 1$ strictly. If $r \ge 1$ or $r \le -1$, the partial sums do not converge to a finite limit and the formula is meaningless.

Three regimes:

Why this matters for wealth planning:

In private-wealth math, $r$ typically represents a discount factor like $1/(1 + i)$where$i > 0$. So$r = 1/1.05 \approx 0.952$for a$5\%$ discount rate. This is always less than 1, so convergence is automatic.

But there is a subtle case where the formula breaks:

If we instead use $r$ as the growth factor $(1 + i)$, then $r > 1$ and the infinite sum diverges. So you need to be careful about which direction the geometric series runs.

The standard convention in CFA materials is:

• For PV of perpetuity: $r$ is the discount factor $< 1$, formula gives $PV = PMT/i$ where $i = (1/r) - 1$
• For FV of perpetuity: divergent — perpetuities have no finite FV

Practical example:

A $1,000 perpetuity discounted at $5\%$ has $PV = \$1{,}000 / 0.05 = \$20{,}000$. This is the finite PV of an infinite payment stream — the further-out payments contribute less and less because they're discounted heavily.

Conversely, the FV of receiving $1,000 forever is infinite — even after 100 years you're still adding to the total. So we never compute FV of perpetuities; only PV.

Connection to Gordon growth model:

The Gordon dividend discount model $PV = D_1/(r - g)$ is the same geometric-series formula with:

• $a = D_1$ (next year's dividend)
• "$r$" in the series $= (1 + g)/(1 + r)$ (effective growth-adjusted discount factor)

For convergence, we need $(1 + g)/(1 + r) < 1$, which means $g < r$. If a company's dividend grows faster than the discount rate, the Gordon model breaks down — this is the classic "growth $>$ discount" violation that finance students hit when over-optimistic on growth projections.

Exam fluency:

You should be able to instantly:

• Recognise when a series converges vs. diverges
• Apply $S_\infty = a/(1-r)$ for valid $r$
• Connect to perpetuity formulas, Gordon DDM, and dynasty-trust math
• Spot the trap: "expected growth is $10\%$ and discount rate is $8\%$" $\to$ Gordon DDM diverges, must use multi-stage DDM instead