**2016 Putnam B1.** Let be the sequence such that and for , (as usual, the function is the natural logarithm.

Show that the infinite series converges and find its sum.

The recurrence relation is begging to be **exponentiated** so that we have on both sides:

and now this looks like a **telescoping series** that allows cancellation of terms:

To show that converges, we have to show that exists. There are many different techniques to show that a limit exists, we can decide which is appropriate by playing around with the sequence.

After working out the first few terms of the sequence, it may become clear that **the sequence is decreasing**. The recurrence relation seems to suggest it as well: if we can state with confidence that is non-negative, then

It seems reasonable that should be positive as well. Let’s try using **induction to prove both statements at once**:

**Induction statement** : For , and .

It is clear that is true. Let us assume that is true for some . To prove , we can work backwards:

which is a true statement for all real numbers ! Hence the sequence is non-negative and we can use the earlier argument to show that . We have shown that , so by induction, the induction statement is true.

**Since we have a decreasing sequence with a fixed lower bound, it means that the sequence converges.** It remains to figure out what the limit is. We can use the recurrence relation to figure that out:

Hence, , and

**Done!**

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