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.