The tutor uses the ratio test to show the infinite series Σ4ⁿ/n! converges.

Example: Check Σ₀^∞4ⁿ/n! for convergence or divergence.

Solution: The ratio test says that, if lim_n→∞|a_n+1/a_n| < 1, then the series converges. In this case, the terms are all positive anyway, so lim_n→∞a_n+1/a_n < 1 will indicate convergence:

a_n+1/a_n = (4ⁿ⁺¹/(n+1)!)/(4ⁿ/n!)

which becomes 4ⁿ⁺¹/(n+1)! * n!/4ⁿ

which simplifies to 4/(n+1).

lim_n→∞4/(n+1) = 0 < 1, so the series Σ₀^∞4ⁿ/n! converges.

Source:

Larson, Roland E. and Robert P. Hostetler. Calculus, third ed. Toronto: D C Heath and Company, 1989.

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