Question

Suppose that a series \(\sum {{a_n}} \)has positive terms and its partial sums

\({s_n}\)satisfy the inequality \({s_n} \le 1000\) for all \(n\). Explain why \(\sum {{a_n}} \)must be convergent.

Short answer

\(\sum {{a_n}} \)is aconvergent series

Step-by-step solution

Definition of Convergent Series

A Series is said to be convergent only if the sequence of the partial sum tends

to a limit.

Given parameters

The given Series \(\sum {{a_n}} \)has positive values and the partial sum \({s_n} \le 1000\).We need to

prove that \(\sum {{a_n}} \)is convergent.

Proving that \(\sum {{a_n}} \)is a convergent series

As Given, The Partial sum \({s_n}\)has an increasing sequence.

We know that\({s_n} - {s_{n - 1}} = {a_n} > 0\)for all n

The Sequence \(\{ {s_n}\} \)is bounded as \({s_n} \le 1000\) for all n

The sequence of partial sum\({s_n}\)converges.Any series which tends to a limit is

a convergent series.

Hence\(\sum {{a_n}} \)is convergent.