#### To determine

**To explain:** The series
∑an
must be convergent.

#### Explanation

**Given:**

A series
∑an
has positive terms and its partial sum
sn≤1000 for all n
.

**Definition used:**

If the sequence of the partial sum of the series is convergent then the series is convergent.

**Result used:** Monotonic sequence theorem

Any monotonic and bounded sequence is convergent.

**Calculation:**

The series
∑an
has positive terms.

That is,

sn−sn−1=an>0

Thus, the sequence of the partial sum
{sn}
is increasing.

From the given,
sn≤1000 for all n
.

That is, the sequence
{sn}
is bounded.

Thus, by Result stated above (Monotonic sequence theorem), the sequence of partial sums converges.

Therefore, the series is convergent by the definition.

Hence, the series
∑an
is convergent.