Q. 13 - Calculus

Question

Explain why every convergent series consisting of positive terms is absolutely convergent.

Step-by-Step Solution

Answer

The series consists of positive terms $\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\left|a_k\right|$. It is absolutely convergent.

Step 1. When the series consists of positive terms.

Consider the series $\sum _{k=1}^{\infty }{a}_k$ is convergent consisting of positive terms.

Since, the series consists of positive terms, then $\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\left|a_k\right|$.

Now according to the definition, the series is absolutely convergent, if the series $\sum _{k=1}^{\infty }\left|a_k\right|$ converges.

Step 2. Check whether the series converges.

The given series to be absolutely convergent the series $\sum _{k=1}^{\infty }\left|a_k\right|$ must converge.

As $\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\left|a_k\right|$ and series $\sum _{k=1}^{\infty }{a}_k$ is convergent, then $\sum _{k=1}^{\infty }\left|a_k\right|$ is convergent.

By definition, the series $\sum _{k=1}^{\infty }\left|a_k\right|$ is absolutely convergent.