1. Determine N so that sum_{n=1}^{N} (- 1)^n {1/{2 n + 1}} differs from the sum of the series...

Question:

1. Determine {eq}N {/eq} so that {eq}\sum_{n=1}^{N}(-1)^n\frac{1}{2n+1} {/eq} differs from the sum of the series {eq}\sum_{n=1}^{\infty}(-1)^n\frac{1}{2n+1} {/eq} by less than {eq}\frac{1}{1000} {/eq}.

2. Find the sum of the series {eq}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n!} {/eq} correct to {eq}4 {/eq} decimals.

Alternating series remainder:

The alternating series remainder theorem states that if an convergent alternating series is evaluated at some values of {eq}n {/eq}, then the absolute value of its remainder shall be less than the rule of series evaluated at {eq}n + 1 {/eq}.