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If we just look at the original ratio, it would look as if it converges, because the bottom term will "out pace" the top as n approaches infinity. But if we want to be sure, we can use Ratio Test and l'Hopital's Rule.

Using the ratio test we replace n with n+1 in the original terms and multiply by the reciprocal of the original ratio.

Getting [2(n+1)-1][5^n n!] / [5^(n+1)][(n+1)!][2n-1]

If we simplify we wind up with [2n+1] / [5(n+1)(2n-1)], now we can use l'Hopital's Rule and take the derivative of the top and bottom, which will leave us with a constant on top and an n term on the bottom.

2n+1 / 5(n^2+2n+1) ----> y' = 2/10(n+1)

If we allow n to approach infinity, we clearly see that the ratio will go to zero, which shows convergence.

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