MATH HELP, infinite series

Question

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

∑sqrt(n)/(n+4)



from n = 1 to infinity

Answer 1

The Series diverges.

The is a very useful Theorem in the theory of series

THEOREM

If the series ∑_n=1^∞ a_n converges , then Lim a_n =0 .

The contrapositive of the above statement is also true. That is

THEOREM

If the Lim a_n ≠ 0 , then the series ∑_n=1^∞ a_n diverges.

In our case of the series in question

∑_n=1^∞ √(n/(n+4)) , and a_n = √(n/(n+4)) and lim a_n = 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