Estimating the Value of a Series — Question 6

*Question 6 The Leibniz series is $\displaystyle\frac\pi4=\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}$. If exactly $M$ terms are used (indices $0$ through $M-1$), find the least $M$ certified by a next-term bound strictly below $10^{-4}$.

*Question 6 – Solution

With $M$ terms, the final included index is $M-1$, so the first omitted term has index $M$ and magnitude $1/(2M+1)$.

$$|R|\le\frac1{2M+1}.$$ We need $$\frac1{2M+1}<10^{-4}\iff 2M+1>10{,}000\iff M>4999.5.$$ Thus the least integer is $\boxed{M=5000}$.

With $4999$ terms the bound is $1/9999>10^{-4}$; with $5000$ terms it is $1/10001<10^{-4}$. This large count illustrates the slow convergence of the Leibniz series.