show if the given series converges or diverges Sum(Upper^infinity,lower n=1) 10^n/n!

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The ratio test says that if the limit `L=lim_{n->infty}|a_{n+1}/a_n|<1` then the series converges.

In this case, `sum_{n=1}^infty 10^n/{n!}` , so `a_n=10^n/{n!}` .  This means the limit is

`L=lim_{n->infty}|10^{n+1}/{(n+1)!} cdot {n!}/{10^n}|`    simplify

`=lim_{n->infty}|10/{n+1}|`

`=0`

Since `L=0<1` the series converges.

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The ratio test says that if the limit `L=lim_{n->infty}|a_{n+1}/a_n|<1` then the series converges.

In this case, `sum_{n=1}^infty 10^n/{n!}` , so `a_n=10^n/{n!}` .  This means the limit is

`L=lim_{n->infty}|10^{n+1}/{(n+1)!} cdot {n!}/{10^n}|`    simplify

`=lim_{n->infty}|10/{n+1}|`

`=0`

Since `L=0<1` the series converges.

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