Identify whether the infinite series converge absolutely conditionally or dont sum_(n=1)^oo (-1)^(n+1) (n (arctan(n+1)-arctan (n))∞∑n=1(−1)n+1(n(arctan(n+1)−arctan(n)) (Apply Mean Value theorem to conclude)?
1 Answer
The series:
is conditionally convergent but not absolutely convergent.
Explanation:
Let
The Mean Value Theorem states that if
Let
with
and multiplying by
Now the series:
is not convergent as we can see using the limit comparison test with the harmonic series. In fact:
By direct comparison we can then conclude that also:
is not convergent hence the series in analysis is not absolutely convergent.
On the other hand, let:
We have that:
and:
hence:
hence
Based on Leibniz' theorem then the series:
are both convergent, then by the squeeze theorem also:
is convergent.