
Yinyan W.
asked 08/10/21MATH HELP, infinite series
serie de n=1 a infinity}(\sqrt{n de)/(n+4)
∑sqrt(n)/(n+4)
from n = 1 to infinity
1 Expert Answer
The Series diverges.
The is a very useful Theorem in the theory of series
THEOREM
If the series ∑n=1∞ an converges , then Lim an =0 .
The contrapositive of the above statement is also true. That is
THEOREM
If the Lim an ≠ 0 , then the series ∑n=1∞ an diverges.
In our case of the series in question
∑n=1∞ √(n/(n+4)) , and an = √(n/(n+4)) and lim an = lim √(n/(n+4)) = { n / (n+1) }1/2 = { 1/ ( 1+ 1/n)}1/2 =1
Since the limit is different than zero according to the contrapositive of the original Theorem
the series diverges
Yinyan W.
thank you!08/11/21
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Bradford T.
What is to be done? Evaluate? Find if converges or diverges?08/10/21