
Infinite Sequences and Series 17-21
Consider Σ Σv
n
n
=
1
, which is a geometric series, with common ratio
r = ∈
1
0 1( , ).
Since Σ Σv
n
n
=
1
is convergent, by comparison test, S
1
n
is also convergent.
Example 17.27 Test for convergence
n
n
2
3 2
1
−
∑
=
∞
.
Solution: Let u
n
n
=
−
>
2
3 2
0
for all n. This is a series of positive terms.
We know that
n
n
n
2
3 2
2
3
1
−
> =
as n gets larger.
Consider
Σ Σv
n
n
=
1
.
It is seen that the above series is harmonic series and we proved earlier that it diverges.
Therefore, by comparison test,
n
n
2
3 2
1
−
∑
=
∞
is also divergent.
Example 17.28 Test for convergence
1
2 10
3 2
1
n n
n
− +
∑
=
∞
.
Solution: Let u
n n
n
=
− +
>
1
2