A Sum Of Zeroes

Calculus Level 1

0πcosxdx=0π2πcosxdx=02π3πcosxdx=03π4πcosxdx=0 \begin{aligned} \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{aligned}

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

0cosxdx=0πcosxdx+π2πcosxdx+2π3πcosxdx+=0+0+0+=0  . \begin{aligned} \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{aligned}

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