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