Is the series #\sum_(n=1)^\infty\tan^1(1/n)# absolutely convergent, conditionally convergent or divergent?
(Use the appropriate test.)
(Use the appropriate test.)
2 Answers
Answer
Answer:
Explanation
Explanation:
Answer:
Diverges by Limit Comparison Test.
Explanation:
The Integral Test can be used; however, this is a very ugly function to integrate.
So, let's use the Limit Comparison Test instead, which tells us if we have some sequence
And if
Let's compare the given series to
So, let's use the Limit Comparison Test:
This is an indeterminate form and we need to use l'Hospital's Rule.
Temporarily rewrite in terms of
First, let's consider
In general, the derivative of the arctangent function is
So, for our function, using the Chain Rule, we'll temporarily say that
Thus,
Thus,
Answer
Answer:
Explanation
Explanation:
Describe your changes (optional) 200
Answer:
Divergent
Explanation:
Setting it up for IBP (with the parenthesized hint):
Wellknown:

# d(tan^(1) u) = 1/(1+ u^2) \du# 
#du = d(1/x) =  1/x^2\ dx#
The second term clearly tends toward
For the first term, we can apply L'Hôpital because:
Applying L'H:
The integral diverges: and if the integral diverges, then the series does so as well.
Describe your changes (optional) 200