Teaching Notes · Math for Social Sciences

Sequences, Series, and the Mathematics of Saving

This last set of notes steps back from derivatives entirely and asks a simpler question: what happens when a fixed procedure, a deposit, a payment, a fixed rate of depreciation, repeats month after month or year after year? The answer runs through sequences and series, and it closes the course on its most directly practical note, since a savings account, a loan, and a depreciating asset all turn out to be the same piece of mathematics wearing three different names.

What is a sequence?

A sequence is simply a function whose domain is the positive integers, producing an ordered list of terms a1, a2, a3, and so on. For a_n = 2n + 1, the first four terms are 3, 5, 7, and 9. For b_n = 3/(n + 1), the first four terms are 3/2, 1, 3/4, and 3/5. A sequence converges if its terms settle toward a single limiting value as n grows without bound, and diverges if they do not. The sequence 1/2, 1/4, 1/8, 1/16, and so on, clearly approaches 0, so lim(n→∞) 1/2ⁿ = 0; this is the same limit idea from earlier in the course, just applied to a list of numbers indexed by whole numbers rather than to a continuous function.

Finding the limit of a more complicated sequence usually means dividing through by the highest power of n present, the same technique used earlier for limits of rational functions as x grows without bound. For a_n = 2n/(n + 1), dividing numerator and denominator by n gives 2/(1 + 1/n), which approaches 2/(1 + 0) = 2. For a_n = n/(1 − 2n), the same technique gives a limit of −1/2. For a_n = 2ⁿ/(2ⁿ − 1), dividing through by 2ⁿ gives 1/(1 − 1/2ⁿ), which approaches 1/(1 − 0) = 1.

A savings account, one month at a time

A deposit of 1000 is placed in an account earning 6 percent annual interest, compounded monthly. After one month, the balance is A1 = 1000 + 1000(0.06/12) = 1000(1.005). After two months, the previous balance earns interest again, giving A2 = 1000(1.005) + 1000(1.005)(0.005) = 1000(1.005)². Continuing this pattern, the balance after n months is A_n = 1000(1.005)ⁿ, and the sequence of monthly balances, 1000(1.005), 1000(1.005)², 1000(1.005)³, and so on, is literally the compound interest formula from the previous set of notes, written out one term at a time rather than as a single closed form equation.

From a sequence to a series

A series is what results from adding the terms of a sequence together rather than just listing them. A finite sum a1 + a2 + ... + a_N is written compactly with sigma notation, ∑_i=1^N a_i, and an infinite series, ∑_n=1^∞ a_n, extends the same idea to adding infinitely many terms. Not every infinite series settles on a finite total; many simply grow without bound. A quick first check, called the nth term test for divergence, states that if the individual terms a_n do not approach 0 as n grows large, the series cannot possibly converge. For ∑ 2ⁿ, the terms themselves grow toward infinity rather than toward 0, so the series diverges immediately. For ∑ n/(1 + 2n), the terms approach 1/2 rather than 0, so this series diverges as well, even though its individual terms stay bounded rather than blowing up.

The geometric series

One particular kind of series, where each term is a fixed multiple of the one before it, turns out to model almost every repeating financial transaction in this course. A geometric series has the form ∑_n=0^∞ arⁿ, where a is the first term and r is the common ratio between consecutive terms. Its partial sum through the nth term has a closed form, s_n = a(1 − rⁿ⁺¹)/(1 − r), for r not equal to 1. For the series ∑ 3(1/4)ⁿ, with a = 3 and r = 1/4, the third, fifth, and tenth partial sums work out to roughly 3.984, 3.999, and 4.000, closing in on 4 as more terms are added, exactly the behavior that makes an infinite geometric series with a small enough ratio useful in the first place.

A monthly savings plan, added up

The partial sum formula turns a stream of regular deposits into a single number. Suppose 50 is deposited every month for 2 years into an account paying 6 percent compounded monthly. The deposit made in the first month grows, by the end of the 24 month term, to 50(1.005)²⁴; the deposit made in the second month, having one fewer month to earn interest, grows to 50(1.005)²³, and so on down to the final deposit, which earns no interest at all. The total balance is the sum of all 24 of these terms, a geometric series with a = 50(1.005) and r = 1.005, which evaluates, using the partial sum formula, to approximately 1277.96. A monthly savings habit, in other words, is nothing more than a finite geometric series, with the partial sum formula doing in one step what adding 24 separate compound interest calculations by hand would otherwise take.

When an infinite series converges

An infinite geometric series ∑ arⁿ diverges whenever the common ratio satisfies |r| ≥ 1, since the terms never shrink enough for the running total to settle down. But whenever |r| < 1, the series converges to a clean closed form sum:

∑_n=0^∞ arⁿ = a / (1 − r)

For ∑ (−1/2)ⁿ, with a = 1 and r = −1/2, since |r| = 1/2 is less than 1, the series converges to 1/(1 − (−1/2)) = 2/3. For ∑ (3/2)ⁿ, the ratio is 3/2, which is greater than 1, so the series diverges. For ∑ 4/3ⁿ, rewritten as ∑ 4(1/3)ⁿ starting from n = 1, the series converges, and summing it (adjusting for the starting index) gives a total of 2.

Depreciation as a geometric series

The same convergence idea describes an asset, or a batch of units, wearing out at a steady percentage rate rather than growing. A manufacturer sells 10,000 units of a product each year, and each unit in use has a 10 percent chance of breaking within any given year, so only 90 percent of the previous year's units remain in use the year after. By the end of year 1, the number of units still in use is 10,000 + 10,000(0.90); by the end of year 2, it is 10,000 + 10,000(0.90) + 10,000(0.90)², and so on, another geometric series, this time with common ratio 0.90. Summing this series through 20 years gives a total of approximately 89,000 units still in use, the mathematics of accumulating replacement sales working exactly the same way whether the quantity involved is money gaining value or a product gradually failing.

Across all three applications in this section, a monthly deposit, a stream of savings, and a depreciating batch of goods, the underlying object is always the same geometric series, with only the first term and the common ratio changing to match whatever is actually being modeled. That single idea, more than any other in this course, is the clearest argument for why a social science student benefits from calculus: it is the same formula, reused.

Source: lecture notes for an introductory calculus course for the social sciences, covering sequences, the limit of a sequence, pattern recognition for sequences, series and convergence, the nth term test for divergence, the geometric series, and the convergence of an infinite geometric series.