Writing a Sum in Summation Notation

Express each sum using summation notation:

(a) \(1^{2}+2^{2}+3^{2}+\cdots+9^{2}\)

(b) \(1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots+\dfrac{1}{2^{n-1}}\)

Solution (a) The sum \(1^{2}+2^{2}+3^{2}+\cdots+9^{2}\) has \(9\) terms, each of the form \(k^{2}\); it starts at \(k=1\) and ends at \(k=9\) : \[ 1^{2}+2^{2}+3^{2}+\cdots+9^{2}=\sum\limits_{k=1}^{9}k^{2} \]

(b) The sum \[ 1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots+\dfrac{1}{2^{n-1}} \] has \(n\) terms, each of the form \(\dfrac{1}{2^{k-1}}\). It starts at \(k=1\) and ends at \(k=n\). \[ 1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots+\dfrac{1}{2^{n-1}} =\sum\limits_{k=1}^{n}\dfrac{1}{2^{k-1}} \]