Define , then
If exists, it would indicate:
(2) Series , converges, since .
The first term converges since it is . The second term is .
For the second term to converge, it requires (limit comparison test against the p-series), so , noting . This also confirms does exist.
This method also applies to limit (1).
It is clear that and that for all . Therefore is decreasing, bounded below and converges to , say.
Since follows from the recurrence we must have . Thus .
Now implies that .
Now consider . By Taylor's theorem, and so
Thus . By the Stolz-Cesaro theorem, , i.e. . Thus .
By the way, the limit can be obtained from l'Hopitals rule. Since we only need to find . Thus
Define the sequence by and Assuming the limit exists, evaluate , and find how the initial value of influences of the value of the limit.
Actually it can be found that the limit also exists except for some initial values of . If the limit does exist, then it is either or for , depending on the initial value of , even when is a complex number.