Determining Whether a Series Converges or Diverges

Determine whether the series \(\sum\limits_{k\,=\,1}^{\infty }k=1+2+3+\cdots\) converges or diverges.

Solution The sequence \(\{S_{n}\}\) of partial sums is \[ \begin{array}{c@{}l} S_{1} & =1 \\ S_{2} & =1+2 \\ S_{3} & =1+2+3 \\ \vdots & \\ S_{n} & =1+2+3+\cdots +n \end{array} \]

557

To express \(S_{n}\) in a way that will make it easy to find \(\lim\limits_{n\,\rightarrow \,\infty }S_{n},\) we use the formula for the sum of the first \(n\) integers: \[ S_{n}=\sum\limits_{k=1}^{n}k=1+2+3+ \cdots +n=\dfrac{n(n+1) }{2} \]

Since \(\lim\limits_{n\,\rightarrow \,\infty }S_{n}=\lim\limits_{n\,\rightarrow \,\infty }\dfrac{n(n+1) }{2} =\infty ,\) the sequence \(\{S_{n}\}\) of partial sums diverges. So, the series \(\sum\limits_{k\,=\,1}^{\infty }k\) diverges.