Check for convergence or divergence in the following sequences?
just tell me which test to use
Currently both come up inconclusive (#L=1# ) with the Root Test ?
#a_n=(1+3/n)^(4n)#
#a_n=(n/(n+3))^n#
just tell me which test to use
Currently both come up inconclusive (
#a_n=(1+3/n)^(4n)# #a_n=(n/(n+3))^n#
1 Answer
A) Converges to
Explanation:
The Ratio and Root Tests are used for determining the behavior of infinite series rather than infinite sequences. Here, they won't really be of use.
Checking the convergence or divergence of a sequence is much simpler, and only requires taking the limit to infinity of the sequence.
If
A) Here, if we take the limit, we see
As a result, we're going to want to use l'Hospital's Rule.
But sequences are not differentiable, so we'll rewrite an equation
Recall the logarithm exponent property, which tells us that
Simplify the argument of the logarithm:
Recall the logarithm quotient property, which tells us that
Finally, we want this in rational function form for using l'Hospital's Rule. Multiplying by
Take the limit to infinity:
This is definitely indeterminate, so differentiate the numerator and denominator:
=
So, we get
Now that we know
The sequence converges to
B) This is also indeterminate:
So, proceed as we did in the previous problem:
So
The sequence converges to