# Can one solve a recurrence that contains a function?

I'd like to solve a recurrence, so I've been reading about solving recurrences, and all the ones I've seen solved involve only previous terms of the recurrence, and constants. My recurrence is

$$t(n) = \frac{t(n-1)(1-\ln(n-1))}{\ln n}$$

and if I could get help solving it, I'd be much obliged; but I'd especially like to know if there are any general principles for solving recurrences of this nature (with functions in them).

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## 1 Answer

When you have $t(n)=t(n-1)f(n)$ you can just write $t(n)=t(1)\prod_{i=2}^n f(i)$. In this case I started $i$ at $2$ because trying $n=1$ leads to division by zero. So $t(n)=t(1)\prod_{i=2}^n \frac {1-\ln (i-1)}{\ln i}$

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... and it's quite unlikely that there is any simplification that can be usefully applied to that product. – Robert Israel Aug 16 '12 at 6:17
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