The summation symbol \(\sum\)
The summation symbol is simply a shorthand notation for a sum of many numbers or variables. Let's see some examples first. Example . The sum \[1 + 2 + 3 + \cdots + 10\] can be written as \[\sum_{i=1}^{10}i.\] That is, \[\sum_{i=1}^{10}i = 1 + 2 + 3 + \cdots + 10.\] Here, the symbol \(\sum\) is called the summation symbol, \(i\) is the index . The above expression means we take the sum of \(i\) where the value of \(i\) varies from 1 to 10. □ Remark . The letter \(\sum\) (pronounced sigma ) is an uppercase letter from the Greek alphabet that corresponds to the letter S in the English alphabet. Why S? Because \(\Sigma\) stands for S ummation. □ Let's see some more examples. Example . \[\sum_{k=1}^{10} k^2 = 1^2 + 2^2 + 3^2 + \cdots + 10^2.\] We can use any letter for the index (\(i\), \(k\), etc.). □ Example . \[\sum_{k=1}^{100} 1 = 1 + 1 + \cdots + 1 \text{(adding 1 for100 times)}\] The index needs not appear in the ``body'' of the summation symbol. In this case, the