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MathCalculus

Alternating Series Error Bound: How Many Terms Approximate cos 1?

Count the constant term, bound the first omitted factorial term, and show why five terms meet the 0.00001 tolerance while four do not.

Question

The cosine expansion is

cos⁡θ=1−θ22!+θ44!−θ66!+⋯ .\cos\theta=1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\cdots.

Determine how many terms are needed to approximate cos⁡1\cos 1 with error at most 0.000010.00001.

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Evidence boundary

This solution uses OpenStax §5.5 exercise 303, the expansion of cos θ, θ=1 radian, and an error of at most 0.00001. N counts retained terms starting with the constant 1. Decimal values are rounded displays; the factorial inequalities establish the guarantee.