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Series

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Series

Study Guide

Key Definition

A series is the indicated sum of the terms of a sequence. For example, the series $3 + 7 + 11 + 15 + 19$ is derived from the finite sequence $3, 7, 11, 15, 19$.

Important Notes

A finite series has a limited number of terms, like $3 + 7 + 11 + 15 + 19$.
An infinite series continues indefinitely, such as $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$.
Sigma notation $\sum$ simplifies the expression of series.
The formula for the sum of an arithmetic series is $S_n = \frac{n}{2}(a + l)$.
The formula for the sum of a finite geometric series (r ≠ 1) is $S_n = a \frac{1 - r^n}{1 - r}$. For an infinite geometric series with |r| < 1, the sum converges to $S = \frac{a}{1 - r}$.

Mathematical Notation

$\sum$ is the summation symbol in sigma notation

$+$ represents addition

$a$ is the first term of a series

$l$ is the last term of an arithmetic series

$n$ is the number of terms

$r$ is the common ratio in a geometric series

Remember to use proper notation when solving problems

Why It Works

For an arithmetic series, pairing terms from the start and end shows each pair sums to the same value, leading to the formula S_n = n/2 * (a + l). For a geometric series, multiplying the sum by r and subtracting from the original series causes most terms to cancel, yielding S_n = a (1 - r^n)/(1 - r). In the special case |r| < 1, the infinite series converges to S = a/(1 - r).

Remember

Key formulas: $S_n = \frac{n}{2}(a + l)$ for arithmetic series; $S_n = a \frac{1 - r^n}{1 - r}$ for finite geometric series (r ≠ 1), and $S = \frac{a}{1 - r}$ for infinite series with |r| < 1.

Quick Reference

Arithmetic Series Sum:$S_n = \frac{n}{2}(a + l)$

Geometric Series Sum:Finite: $S_n = a \frac{1 - r^n}{1 - r}$ (r ≠ 1); Infinite (|r| < 1): $S = \frac{a}{1 - r}$

Understanding Series

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Video explanation of this concept

Beginner

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Beginner Explanation

A series is the sum of terms in a sequence, like $3 + 7 + 11 + 15 + 19$.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the series $1 + 2 + 3 + 4 + 5$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Calculate the total distance covered if you walk $2$ km on the first day, $4$ km on the second day, and increase by $2$ km each day for $5$ days.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the series $2 + 4 + 8 + \cdots$. What is the sum of the first $n$ terms?

Challenge Quiz

Single Choice Quiz

Advanced

Find the sum of the series $2 + 4 + 6 + \cdots + 50$.

Recap

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