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8.2 Summation Notation

What is summation notation?

Summation notation is a way of writing a series such as

\[1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots+\frac{1}{n^2}+\cdots\]

in a shorter form. The shorter form captures the fact that it is a sum by using the greek letter capital sigma \(\Sigma\), indicates the index, in this case the letter \(n\), the starting and ending values of the index \(n=1\) and infinity \(\infty\) (to indicate that the index increases one unit at a time without bound) and finally, the formula for the general term \(a_n=\frac{1}{n^2}\).

\[1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots+\frac{1}{n^2}+\cdots=\sum_{n=1}^\infty\frac{1}{n^2}\]

Summation notation is analogous to sequence notation \(\{a_n\}\) with the exception that in sequence notation the sequence usually starts with \(n=1\) and procedes indefinitely, so it is unnecessary to specify that \(n\) goes from \(1\) to infinity.

Exercise 8.2.1

Rewrite the finite series (\(n\) does not go to infinity) using summation notation: \(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\cdots+\dfrac{50}{51}\)

See Solution

Exercise 8.2.2

Calculate the value of the sum:\(\quad\displaystyle\sum_{p=1}^4(p-1)^2\)

See Solution

Exercise 8.2.3

Calculate the value of the sum:\(\quad\displaystyle\sum_{k=3}^7(k^2-2)\)

See Solution

Exercise 8.2.4

Write the following as a single sum:\(\quad\displaystyle\sum_{p=0}^{3}p^2+\sum_{p=4}^{6}p^2\)

See Solution

Exercise 8.2.5

Write the following as a single sum:\(\quad\displaystyle\sum_{p=0}^{3}p^2+\sum_{q=4}^{6}q^2\)

See Solution

Defining functions using summation notation

Suppose \(f(x)=1-x+x^2-x^3\). We could use summation notation and write

\[f(x=\sum_{n=0}^3(-1)^nx^n\]

Take special notice of how the factor \((-1)^n\) causes the sum to alternate in sign.

Exercise 8.2.6

Write the function in summation notation: \(f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-\cdots+\frac{1}{101}x^{100}\)

See Solution