#### To determine

Whether the series is convergent or divergent and obtain the sum if the series is convergent.

#### Answer

The series is divergent.

#### Explanation

**Given:**

The series is
∑k=1∞k2k2−2k+5
.

Here,
ak=k2k2−2k+5
.

**Theorem used:** Series test for Divergence

If
limn→∞an
does not exist or
limn→∞an≠0
, then the series
∑n=1∞an
is divergent.

**Calculation:**

Obtain the limit of the sequence (the value of the term
ak
as *k* tends to infinity).

That is, compute the value of
limk→∞ak=limk→∞k2k2−2k+5
.

Divide the numerator and the denominator by the highest power.

limk→∞k2k2−2k+5=limk→∞k2k2k2−2k+5k2=limk→∞1k2k2−2kk2+5k2

Redefine the terms as given below.

limk→∞k2k2−2k+5=limk→∞11−2k+5k2=limk→∞1limk→∞1−limk→∞2k+limk→∞5k2=11−0+0=1

Thus, the limit of the sequence is
limk→∞k2k2−2k+5=1
.

Since
limk→∞k2k2−2k+5≠0
and by using the Theorem (Series test for Divergence), the series
∑k=1∞k2k2−2k+5
is divergent.

Therefore, the series is divergent.