Q. 17 - Calculus

Question

Let $\left\{a_k\right\}$ be a sequence of positive numbers. Explain why, if the series $\sum _{k=1}^{\infty } a_{2k-1}$ and $\sum _{k=1}^{\infty } a_{2k}$ both converge absolutely, the series $\sum _{k=1}^{\infty } a_k$ converges absolutely.

Step-by-Step Solution

Answer

As the given series is absolutely convergent and $\sum _{k=1}^{\infty } a_{2k-1}$ is the series of odd numbers  and $\sum _{k=1}^{\infty } a_{2k}$is the series of even numbers.

Step 1 . Given information.

Consider the given question,

$\left\{a_k\right\}$ is the sequence of positive numbers.

Step 2. Explain the series.

The series $\sum _{k=1}^{\infty } a_{2k-1}$ is the series of odd numbers $\sum _{k=1}^{\infty } a_{2k}$ is the series of even numbers. Both the series converge absolutely.

The series $\sum _{k=1}^{\infty } a_k$ is the sum of series $\sum _{k=1}^{\infty } a_{2k-1},\sum _{k=1}^{\infty } a_{2k}$.

As the sum of two convergent series is convergent.

Moreover, the series $\sum _{k=1}^{\infty } a_{2k-1},\sum _{k=1}^{\infty } a_{2k}$is absolutely convergent.

Therefore, the series $\sum _{k=1}^{\infty } a_k$ is convergent.