Summation Definition

Summation

We use the notations

${\sum }_{j=m}^n{a}_j\text{or}{\sum }_{j=m}^n{a}_j\text{or}{\sum }_{1\le j\le n}^{}{a}_j$

to represent $a_m+a_{m+1}+\dots +a_n$ where:

• $j$ = index
• $m$ = lower limit
• $n$ = upper limit
• $a_j$ = summand

Example

Express the sum of the first 100 elements of the sequence $\{a_n\}$ where $a_n=1$ for $n=1,2,3,\dots$

lower limit = $1$, upper limit = $100$, $⟹{\sum }_{j=1}^{100}\frac{1}{j}$

Question

What is the value of ${\sum }_{j=1}^5{j}^2$?

Answer

$$\begin{array}{rl}& \sum _{j=1}^5{j}^2=1^2+2^2+3^2+4^2+5^2\\ & =1+4+9+16+25\\ & =55\end{array}$$

Question

What is the value of ${\sum }_{k=4}^8(-1{)}^k$?

Answer

$$\begin{array}{rl}& \sum _{k=4}^8(-1{)}^k=(-1{)}^4+(-1{)}^5+(-1{)}^6+(-1{)}^7+(-1{)}^8\\ & =1+(-1)+1+(-1)+1\\ & =1\end{array}$$