Determine if the functions converge or diverge?I dont know how to do these! Please help by explaining so I can understand! Thanks! Example:...

A sries like u1+u2+u3+.....un+..... is  said to be convergent if its partial sum Sn = u1+u2+u3+u4+....un  has a finite limit as n approaches infinity.

A necessary condtion for this is that the nth term should have limit 0.

In this case the nth term un = (5n^4)+1/((150,348n^3)+999) is

Lt (5n^4) +Lt (1/((150,348n^3)+999) as n--> inf.

= inf+0 = inf. So the series diverges;

Even if  you write the nth term as (5n^4+1)/((150,348n^3)+999). The limit of this behave like  5n^4/150348n^3 as n-->infinity. Or like

(5/150348) n which approaches infinity as n--> infinity.

So the series diverge.

b)

To detrmine the nature of (convergent or otherwise) of the series 2, 1, 2/3, 2/5.... The series is rewritten as:

2/1,   2/2, 2/3,  2/4,  2/5,   2/6,........2/n.......... Or

So the nth term of this series is 2/n. Which could be compared with popular series 1, 1/2, 1/3,1/4,1/5, etc. which is divergent in the sense Su(1/n) for n=1 to inf approach infinite.

Since each term of the given series is 2 times the latter, the former series also diverges.