# A Sum Of Zeroes

Calculus Level 1

$\begin{eqnarray} \displaystyle \int_0^{\pi } \cos x \, dx &= & 0 \\ \displaystyle \int_{\pi}^{2\pi} \cos x \, dx &= & 0 \\ \displaystyle \int_{2\pi}^{3\pi} \cos x \, dx &= & 0 \\ \displaystyle \int_{3\pi}^{4\pi} \cos x \, dx &= & 0 \\ &\vdots & \end{eqnarray}$

Because all the equation above are true, is the following equation true as well?

$\begin{eqnarray} \int_0^\infty \cos x \, dx &= &\int_0^\pi \cos x \, dx + \int_{\pi}^{2\pi} \cos x \, dx + \int_{2\pi}^{3\pi} \cos x \, dx + \cdots \\ &=& 0+ 0+0+ \cdots \\ &=& 0 \; . \end{eqnarray}$

×