Advanced High-School Mathematics

SECTION 5.2 Numerical Series 271

nlim→∞

(

√^1

n+ 1

)

( 1

n

) =∞,

showing that the terms of the series

∑∞

n=0

1

√

n+ 1

are asymptotically

much bigger than the terms of the already divergent series

∑∞

n=1

1

n

. There-

fore, by the Limit Comparison Test,

∑∞

n=0

√^1

n+ 1

diverges.

Example 3. The series

∑∞

n=1

n^2 + 2n+ 3

n^9 /^2

converges. We compare it

with the convergent series

∑∞

n=1

1

n^2

:

nlim→∞

Ñ

n^2 + 2n+ 3

n^9 /^2

é

(

1

n^2

) = limn→∞n

(^2) + 2n+ 3
n^7 /^2

= 0,

proving convergence.

Example 4. The series

∑∞

n=2

1

(lnn)^2

diverges. Watch this: