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.
Rewrite the finite series (\(n\) does not go to infinity) using summation notation: \(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+\cdots+\dfrac{50}{51}\)
Calculate the value of the sum:\(\quad\displaystyle\sum_{p=1}^4(p-1)^2\)
Calculate the value of the sum:\(\quad\displaystyle\sum_{k=3}^7(k^2-2)\)
Write the following as a single sum:\(\quad\displaystyle\sum_{p=0}^{3}p^2+\sum_{p=4}^{6}p^2\)
Write the following as a single sum:\(\quad\displaystyle\sum_{p=0}^{3}p^2+\sum_{q=4}^{6}q^2\)
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.
Write the function in summation notation: \(f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-\cdots+\frac{1}{101}x^{100}\)