9.2 Series and Summation Notation

Series and summation notation give you a compact way to represent and calculate the sum of many terms at once. Instead of writing out dozens (or infinitely many) terms, you can express the entire sum in a single expression. This section covers the two main types of series, how sigma notation works, the key formulas for finding sums, and where these ideas show up in real-world problems.

Arithmetic vs Geometric Series

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Defining Arithmetic and Geometric Series

A series is the sum of the terms of a sequence. The two types you'll work with most are arithmetic and geometric.

• An arithmetic series is the sum of terms where each consecutive term differs by a constant amount called the common difference ($d$). The terms grow or shrink by the same amount each time.
• A geometric series is the sum of terms where each term is found by multiplying the previous term by a fixed, non-zero value called the common ratio ($r$). The terms grow or shrink by the same factor each time.

The distinction matters because each type has its own sum formula. Mixing them up is a common mistake on tests.

Representing Arithmetic and Geometric Series

Both types can be described with recursive or explicit formulas. Recursive formulas define each term using the previous term. Explicit formulas let you jump straight to any term based on its position.

Arithmetic series:

• Recursive: $a_n = a_{n-1} + d$
• Explicit: $a_n = a_1 + (n-1)d$

Geometric series:

• Recursive: $a_n = a_{n-1} \cdot r$
• Explicit: $a_n = a_1 \cdot r^{n-1}$

In both cases, $a_1$ is the first term and $n$ is the term's position. The explicit formula is usually more useful for series problems because you can find any term without computing all the ones before it.

Summation Notation for Series

Introducing Summation Notation

Summation notation uses the Greek letter sigma ($\Sigma$) to express a sum compactly. The general form is:

$\sum_{i=m}^{n} a_i$

This tells you to evaluate $a_i$ for every integer value of $i$ from $m$ to $n$, then add all those results together.

Components of Summation Notation

• Index of summation ($i$): the variable that changes with each term. It counts from the lower limit up to the upper limit.
• Lower limit ($m$): the starting value of $i$.
• Upper limit ($n$): the ending value of $i$.
• General term ($a_i$): the expression you evaluate for each value of $i$.

To evaluate a summation, plug in each integer value of $i$ from $m$ to $n$ into the general term, then add everything up. For example, $\sum_{i=1}^{4} 2i$ means $2(1) + 2(2) + 2(3) + 2(4) = 2 + 4 + 6 + 8 = 20$.

Calculating Series Sums

Sum of an Arithmetic Series

The sum of the first $n$ terms of an arithmetic series is:

$S_n = \frac{n}{2}[2a_1 + (n-1)d]$

There's an equivalent form that's sometimes easier to use when you already know the last term ($a_n$):

$S_n = \frac{n}{2}(a_1 + a_n)$

This second version is just the average of the first and last terms, multiplied by the number of terms. It's worth remembering both.

Sum of a Geometric Series

Finite geometric series (any value of $r \neq 1$):

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

Infinite geometric series (only when $|r| < 1$):

$S_{\infty} = \frac{a_1}{1 - r}$

The infinite formula only works when the absolute value of the common ratio is less than 1. That condition guarantees the terms shrink toward zero, so the sum converges to a finite number. If $|r| \geq 1$, the infinite series diverges and has no finite sum.

Solving for Series Sums

When you're given a series sum problem, work through these steps:

1. Identify the type of series. Check whether consecutive terms share a common difference (arithmetic) or a common ratio (geometric).
2. Find the key values. Determine $a_1$, $n$, and either $d$ or $r$.
3. Choose the right formula. Use the arithmetic sum formula if there's a common difference, or the geometric sum formula if there's a common ratio. For an infinite geometric series, confirm $|r| < 1$ before using $S_{\infty}$.
4. Substitute and solve. Plug your values into the formula and simplify.

A quick check: for arithmetic series, the sum should be close to $n$ times the average term. For geometric series with $|r| < 1$, the infinite sum should be larger than $a_1$ but not wildly so.

Real-World Applications of Series

Modeling with Arithmetic Series

Arithmetic series model situations where a quantity increases or decreases by a fixed amount each period:

• Simple interest: You earn the same dollar amount of interest every period. If you earn $50 per year on a $1,000 deposit, the total interest after 10 years is the sum $50 + 50 + 50 + \cdots$ (10 terms) = $500.
• Salary increases: An employee gets a fixed annual raise. A starting salary of $40,000 with a $2,000 raise each year forms an arithmetic series.
• Straight-line depreciation: An asset loses the same dollar value each year. A $20,000 car that loses $3,000 per year follows an arithmetic pattern.

Modeling with Geometric Series

Geometric series model situations where a quantity grows or shrinks by a fixed percentage each period:

• Compound interest: Each period's interest is a percentage of the current balance, so the balance grows by a constant ratio. A $1,000 investment at 5% annual interest has a ratio of $r = 1.05$.
• Population growth: A bacteria colony that doubles every hour has $r = 2$. After $n$ hours, the total number of bacteria produced is a geometric series.
• Radioactive decay: A substance loses a fixed fraction of its mass each half-life ($r < 1$), so the remaining amounts form a geometric series that converges.

Solving Real-World Problems

1. Read the problem and identify whether the growth/decline is by a constant amount (arithmetic) or a constant percentage (geometric).
2. Extract $a_1$, $d$ or $r$, and $n$ from the context.
3. Apply the appropriate sum formula.
4. Interpret your answer in context. Units matter: if you're summing dollars, your answer is in dollars. If you're summing population counts, say so.