Math Help Infinite Series

serie de n=1 a infinity}(sqrt{n de)/(n+4)

∑sqrt(n)/(n+4)



from n = 1 to infinity

Solution

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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