Question

If \(\sum {{a_n}} \)and \(\sum {{b_n}} \)both aredivergent, is \(\sum {\left( {{a_n} + {b_n}} \right)} \) necessarilydivergent?

Short answer

\(\sum {\left( {{a_n} + {b_n}} \right)} \)is not necessarily a divergent series.

Step-by-step solution

Definition of Convergent and Divergent Series

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

to a limit.A Series is said to be divergent only if the sequence of the partial sum

does not reach a limit and it will be usually∞

Given parameters

The given Series \(\sum {{a_n}} \)is divergentand \(\sum {{b_n}} \) is divergent.We need to checkif

\(\sum {\left( {{a_n} + {b_n}} \right)} \)is a divergent series or not?

Proving that \(\sum {\left( {{a_n} + {b_n}} \right)} \)is a not necessarily a divergent series

Considering that\(\sum\limits_{n = 1}^\infty {{a_n}} \) is divergent and \(\sum\limits_{n = 1}^\infty {{b_n}} \) is divergent

Let us assume that

\(\sum {{a_n}} = n\)-- Equation 1

\(\sum {{b_n}} = n\)-- Equation 2

By Adding Equation 1 and Equation 2 we will get

\(\sum {\left( {{a_n} + {b_n}} \right)} = \sum {n + \sum { - n} } \)

\(\sum {\left( {{a_n} + {b_n}} \right)} = \sum {\left( {n - n} \right) = \sum 0 } \)

Any series which tends to a limit is a convergent series. The given series

converges with sum 0

Hence \(\sum {\left( {{a_n} + {b_n}} \right)} \)is not necessarily a divergent series