Answer:
Wow this hard
Step-by-step explanation:
X∞
k=1
k(k + 2)
(k + 3)2
is convergent or divergent. If it is convergent, find its sum.
Answer: This series diverges. To see this, I will show that the terms in the sequence do not
go to zero:
lim
k→∞
k(k + 2)
(k + 3)2
= lim
k→∞
k
2 + 2k
k
2 + 6k + 9
.
Dividing numerator and denominator by k
2 yields
lim
k→∞
1
k
2
1 + 2
k
1
k
2
1 + 6
k +
9
k
2
= lim
k→∞
1 + 2
k
1 + 6
k +
9
k
2
= 1.
Therefore, using the nth term test (a.k.a. Test for Divergence), the series diverges
