Define the sequence by and Evaluate

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- July 29th 2009, 09:06 PMNonCommAlgLimit (5)
Define the sequence by and Evaluate

- July 30th 2009, 05:41 AMDeMathNon-rigorous proff (solution)
- August 6th 2009, 06:32 PMluobo
Define , then

If exists, it would indicate:

(1)

(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).

- August 8th 2009, 06:05 AMhalbard
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

. - August 12th 2009, 05:24 AMluobo
- August 12th 2009, 06:43 AMluobo
It would be interesting to investigate when is defined in complex domain, i.e.

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.