#### To determine

Whether the series is convergent or divergent.

#### Answer

The series is convergent.

#### Explanation

**Result used**:

The
p
-series
∑n=1∞1np
is convergent if
p>1
and divergent if
p≤1
.

**Given:**

The series is
∑n=1∞n+4n2
.

**Calculation:**

The given series can be expressed as follows,

∑n=1∞n+4n2=∑n=1∞nn2+∑n=1∞4n2=∑n=1∞n12n2+∑n=1∞4n2=∑n=1∞1n2−12+∑n=1∞4n2

=∑n=1∞1n3/2+∑n=1∞4n2
(1)

Consider the series
∑n=1∞1n3/2
.

Clearly, this is *p-*series with
p=3/2
and
p>1
.

Use the Result stated above, the series
∑n=1∞1n3/2
is convergent.

Consider the series
∑n=1∞1n2
.

Clearly, this is *p-*series with
p=2 and p>1
.

Use the Result stated above, the series
∑n=1∞1n2
is convergent.

Note that, a nonzero multiple of a convergent series is also convergent.

Therefore, the series
4∑n=1∞1n2
is convergent.

Note that, the sum of two convergent series is convergent.

From the equation (1), the series
∑n=1∞1n3/2+∑n=1∞4n2
is convergent.

Therefore, the given series
∑n=1∞n+4n2
is convergent.