Question

Given two infinite series \(\sum a_{n}\) and \(\sum b_{n}\) such that \(\sum a_{n}\) converges and \(\sum b_{n}\) diverges, prove that \(\sum\left(a_{n}+b_{n}\right)\) diverges.

Short answer

The series \(\sum\left(a_{n}+b_{n}\right)\) diverges because it is a sum of a divergent series and a convergent series.

Step-by-step solution

Understand what convergent and divergent series are

A series \(\sum a_{n}\) is said to converge if the sequence of its partial sums \(\{s_{n}\}\), defined by \(s_{n} = a_{1} + a_{2} + \ldots + a_{n}\), approaches a limit as \(n\) approaches infinity. A series is said to diverge if it is not convergent.

Evaluate the given data

We are given that the series \(\sum a_{n}\) is convergent and the series \(\sum b_{n}\) is divergent.

Sum of the series

The series \(\sum\left(a_{n}+b_{n}\right)\) can be rewritten as \(\sum a_{n} + \sum b_{n}\).

Apply the understanding of convergent and divergent series

According to the properties of convergent series, the sum of a divergent series and a convergent series is a divergent series.

Conclusion

Therefore, the given series \(\sum\left(a_{n}+b_{n}\right)\) diverges.