A calculus problem by Christopher Boo

Calculus Level 1

True or False?

Let \[\begin{align} S &= \color{Blue}{\frac{1}{1}} \color{Red}{-\frac{1}{2}} \color{Blue}{+\frac{1}{3}} \color{Green}{-\frac{1}{4}} \color{Blue}{+\frac{1}{5}} {\color{Red}{- \frac{1}{6}}} \color{black}+ \cdots \\ \Rightarrow \frac{1}{2}S &= \color{Red}{\frac{1}{2}} \color{Green}{- \frac{1}{4}} \color{Red}{+ \frac{1}{6}} \color{Green}{- \frac{1}{8}} \color{black}+ \cdots. \end{align}\] Adding the two equations and regrouping gives \[\begin{align} \frac{3}{2} S &= \color{Blue}{\bigg ( \frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \cdots \bigg )} - \color{Green}{2 \bigg ( \frac{1}{4} + \frac{1}{8} + \frac{1}{12} + \cdots \bigg )} \\ &= \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = S \\\\ \Rightarrow S &= 0. \end{align}\] Therefore, \[\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = 0.\]

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