13.4 Series and Their Notations

Series Notation and Formulas

A series is what you get when you add up the terms of a sequence. While a sequence is just a list of numbers, a series is their sum. Series notation gives you a compact way to write these sums, and specific formulas let you calculate them without adding every single term by hand.

Summation Notation

Summation notation uses the Greek letter sigma ($\sum$) to represent the sum of a series of terms. Here's how to read it:

• The lower limit (written below $\sum$) tells you where to start counting
• The upper limit (written above $\sum$) tells you where to stop
• The expression after $\sum$ is the formula that generates each term

For example, $\sum_{i=1}^{5} 2i$ means "plug in $i = 1, 2, 3, 4, 5$ into $2i$ and add the results," giving you $2 + 4 + 6 + 8 + 10 = 30$.

An arithmetic series in sigma notation looks like: $\sum_{i=1}^{n} (a + (i-1)d)$, where $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

A geometric series in sigma notation looks like: $\sum_{i=0}^{n-1} ar^i$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Arithmetic Series Sum Calculation

The sum of a finite arithmetic series is:

$S_n = \frac{n}{2}(2a + (n-1)d)$

• $S_n$ = sum of the first $n$ terms
• $a$ = first term
• $d$ = common difference between consecutive terms
• $n$ = number of terms

An equivalent form you'll sometimes see is $S_n = \frac{n}{2}(a_1 + a_n)$, which just says the sum equals the number of terms times the average of the first and last terms. Both formulas give the same answer.

Example: Find the sum of the first 10 terms of an arithmetic series with $a = 2$ and $d = 3$.

1. Identify your values: $a = 2$, $d = 3$, $n = 10$
2. Plug into the formula: $S_{10} = \frac{10}{2}(2(2) + (10-1)(3))$
3. Simplify inside the parentheses: $= 5(4 + 27)$
4. Calculate: $= 5(31) = 155$

Geometric Series Sum Computation

Finite geometric series:

$S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1$

• $S_n$ = sum of the first $n$ terms
• $a$ = first term
• $r$ = common ratio
• $n$ = number of terms

Infinite geometric series:

$S_\infty = \frac{a}{1 - r}, \quad |r| < 1$

This formula only works when $|r| < 1$, meaning the common ratio is between $-1$ and $1$ (exclusive). That's the condition for convergence: the terms get smaller and smaller, so the sum approaches a finite value. If $|r| \geq 1$, the series diverges, meaning the sum grows without bound and you can't assign it a finite value.

Example: Find the sum of the infinite geometric series with $a = 1$ and $r = \frac{1}{2}$.

1. Check convergence: $|r| = \frac{1}{2} < 1$ ✓
2. Apply the formula: $S_\infty = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$

So the series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ adds up to exactly 2.

Sequences vs. Series

Keep these terms straight:

• A sequence is an ordered list of numbers (e.g., $2, 5, 8, 11, \ldots$)
• A series is the sum of the terms of a sequence (e.g., $2 + 5 + 8 + 11 + \cdots$)
• A series converges if its sum approaches a specific finite number; it diverges if it doesn't
• A recursive formula defines each term using previous terms (e.g., $a_n = a_{n-1} + 3$), as opposed to an explicit formula that calculates any term directly from its position

Applications of Series

Real-World Scenarios

Arithmetic and geometric series show up whenever quantities grow or accumulate in predictable patterns.

Arithmetic series model situations with a constant difference between consecutive terms:

• Making regular deposits into a savings account (e.g., depositing $50 more each month than the previous month)
• Seating in an auditorium where each row has a fixed number of additional seats compared to the row in front

Geometric series model situations with a constant ratio between consecutive terms:

• Population growth or decay (bacteria doubling every hour, radioactive material losing half its mass each year)
• Compound interest on investments or loans

Financial Applications

Compound interest calculates interest on both the initial principal and all previously accumulated interest:

$FV = PV(1 + r)^n$

• $FV$ = future value
• $PV$ = present value (initial principal)
• $r$ = periodic interest rate (as a decimal)
• $n$ = number of compounding periods

Example: Find the future value of $1,000 invested at 5% annual interest, compounded quarterly for 10 years.

1. Convert the annual rate to a quarterly rate: $r = 0.05 / 4 = 0.0125$
2. Find the total number of compounding periods: $n = 4 \times 10 = 40$
3. Plug in: $FV = 1000(1 + 0.0125)^{40} \approx 1{,}643.62$

Annuity (a series of equal payments at regular intervals):

The present value of an annuity tells you how much a stream of future payments is worth right now:

$PV = PMT \cdot \frac{1 - \frac{1}{(1+r)^n}}{r}$

• $PV$ = present value of the annuity
• $PMT$ = payment amount per period
• $r$ = periodic interest rate (as a decimal)
• $n$ = total number of payments

Notice how the annuity formula is built on the same geometric series logic: each payment is discounted by a different power of $(1 + r)$, and the formula sums all those discounted values at once.