# Consider two series sum_(n=1)^oo 1/2pinZ and sum_(n=0)^oo 1/pi^2n. Which o

Consider two series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2}\pi nZ\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\pi }^{2}}n$. Which of them converges?
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pierretteA

Series A: $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2}\pi n=\frac{1}{2}\pi \left\{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots \right\}$
Harmonic series – Divergent
Series B: $\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{\pi }^{2}}n=\left\{\left(\frac{1}{{\pi }^{2}}\right)+{\left(\frac{1}{{\pi }^{2}}\right)}^{2}+\dots \right\}$
Geometric series, so $\frac{1}{{\pi }^{2}}<1$ – Convergent
Thus, only series B converges.