Test the convergence of the following series?
2 Answers
The series:
is convergent for
Explanation:
The general term of the series is:
Using the ratio test we can evaluate the ratio.
Then:
It follows that the series is convergent for
For
Using Stirling's approximation:
we can see that
which means:
As the limit is finite, based on the limit comparison test they have the same character, and the series
is divergent based on the
In conclusion the series:
is convergent for
The series converges for
Explanation:
We are given the first four terms of a series:
We can test the convergence of this series using the ratio test.
Let
Now let's evaluate the limit of the ratio of these two terms as
This is an indeterminate limit with the form
The limit in the exponent is now in the form
Applying L'Hôpital's rule to the exponent (taking the derivative of both the numerator and denominator), we get:
Therefore our ratio test yields the result:
According to the ratio test, the series will converge if:
But the question states that