Let
$a>1$ be an integer. Give a non-integral expression in terms of $a$ for $F(a)=\displaystyle\int_1^a (-1)^{\lfloor x \rfloor} \lfloor x \rfloor^{-1} dx,$ where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x.$
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Sketch the graph
$y = \frac{1}{\lfloor{x}\rfloor}$for$x > 1.$ -
How does multiplying it by
$(-1)^{\lfloor{x}\rfloor}$change the graph? -
An integral of a function is just the area under its graph.
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Could you express the total area as a sum?
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Plotting the function
$y = (-1)^{\lfloor{x}\rfloor}\frac{1}{\lfloor{x}\rfloor}$yields rectangles of width$1$and height$-1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, -\frac{1}{5}, \ldots.$Since an integral of a function is just the area under its graph,$F(a)=\sum_{i=1}^{a-1}\frac{(-1)^i}{i}.$