2.2.1 Sum of Series

The following are special series whose sums should be memorized:

Sum of the First m Integers

$\sum\_{n=1}^{m} n = 1 + 2 + 3 + \cdots + m = \frac{m(m + 1)}{2}$

Example: $1 + 2 + 3 \cdots + 11 = \frac{11 \cdot 12}{2} = 66$

Sum of the First m Odd Integers

$\sum\_{n=1}^{m} (2n - 1) = 1 + 3 + 5 + \cdots + (2m - 1) = \left(\frac{(2m - 1) + 1}{2}\right)^2 = m^2$

Example: $1 + 3 + 5 + \cdots + 15 = \left(\frac{15 + 1}{2}\right)^2 = 8^2 = 64$

Sum of the First m Even Numbers

$\sum\_{n=1}^{m} 2n = 2 + 4 + 6 + \cdots + 2m = m(m + 1)$

Example: $2 + 4 + 6 + \cdots + 22 = \frac{22}{2} \cdot \left(\frac{22}{2} + 1\right) = 11 \cdot 12 = 132$

Sum of First m Squares

$\sum\_{n=1}^{m} n^2 = 1^2 + 2^2 + \cdots + m^2 = \frac{m(m + 1)(2m + 1)}{6}$

Example: $1^2 + 2^2 + \cdots + 10^2 = \frac{10(11)(21)}{6} = 35 \cdot 11 = 385$

Sum of the First m Cubes

$\sum\_{n=1}^{m} n^3 = 1^3 + 2^3 + \cdots + m^3 = \left(\frac{m(m + 1)}{2}\right)^2$

Example: $1^3 + 2^3 + \cdots + 10^3 = \left(\frac{10 \cdot 11}{2}\right)^2 = 55^2 = 3025$

Sum of the First m Alternating Squares

$\sum\_{n=1}^{m} (-1)^{n+1}n^2 = 1^2 - 2^2 + 3^2 - \cdots \pm m^2 = \pm \frac{m(m + 1)}{2}$

Examples: $1^2 - 2^2 + 3^2 - \cdots + 9^2 = \frac{9 \cdot 10}{2} = 45$ $1^2 - 2^2 + 3^2 - \cdots - 12^2 = -\frac{12 \cdot 13}{2} = -78$

Sum of a General Arithmetic Series

$\sum\_{i=1}^{m} a_i = a_1 + a_2 + \cdots + a_m = \frac{(a_1 + a_m) \cdot m}{2}$

To find the number of terms: $m = \frac{a_m - a_1}{d} + 1$ where $d$ is the common difference.

Example: $8 + 11 + 14 + \cdots + 35$ $m = \frac{35 - 8}{3} + 1 = 10$ $\text{Sum} = \frac{(8 + 35) \cdot 10}{2} = 43 \cdot 5 = 215$

Sum of an Infinite Geometric Series

$\sum\_{n=0}^{\infty} a_1 \cdot d^n = a_1(1 + d + d^2 + \cdots) = \frac{a_1}{1 - d}$ Where $|d| < 1$.

Examples: $3 + 1 + \frac{1}{3} + \cdots = \frac{3}{1 - 1/3} = \frac{3}{2/3} = \frac{9}{2}$ $4 - 2 + 1 - \frac{1}{2} + \cdots = \frac{4}{1 - (-1/2)} = \frac{4}{3/2} = \frac{8}{3}$

Special Cases: Factoring

Sometimes simple factoring can lead to an easier calculation.

$3 + 6 + 9 + \cdots + 33 = 3(1 + 2 + \cdots + 11) = 3 \left(\frac{11 \cdot 12}{2}\right) = 198$ $11 + 33 + 55 + \cdots + 99 = 11(1 + 3 + 5 + \cdots + 9) = 11(5^2) = 275$

Word Problems

Example: The sum of three consecutive odd numbers is 129, what is the largest? Represent as $(n-2) + n + (n+2) = 129$. Divide by 3 to get the middle term: $129/3 = 43$. Largest is $43+2=45$.

Example: The sum of four consecutive even numbers is 140, what is the smallest? Divide by 4 to get the average (between 2nd and 3rd term): $140/4 = 35$. Middle terms are 34 and 36. Smallest is 32.

Problem Set 2.2.1