Sequence of Partial Sums[br][br]Given a sequence [math]\{a_n\}[/math], the [b]partial sums[/b] form a new sequence [math]\{S_k\}[/math] defined by[br][br][math]S_k = \sum_{n=1}^{n} a_k = a_1 + a_2 + \cdots + a_k[/math][br][br]The left graph shows the terms [math]a_n[/math] of the sequence. The right graph shows the corresponding partial sums [math]S_k[/math].[br][br]Things to try:[br][br][*]Adjust the slider [b]n[/b] to see how the partial sums accumulate term by term.[br][/*][*]Edit the [b]Sequence formula[/b] input box to explore different sequences.[br][/*][*][b]Example:[/b] [code]2*(1/3)^n[/code] — a convergent geometric series; watch [math]S_n[/math] approach [math]1[/math][br][/*][*][b]Example:[/b] [code](-1)^(n+1)/n[/code] — the alternating harmonic series; converges to [math]\ln 2[/math][br][/*][*][b]Example:[/b] [code]1/n[/code] — the harmonic series; does [math]S_n[/math] converge?[br][/*][*][b]Example:[/b] [code]1/n^2[/code] — converges to [math]\pi^2/6[/math][br][/*][*]Notice how a sequence can have terms approaching [math]0[/math] yet still diverge.[/*]