Q. 16 - Calculus

Question

Fill in the blanks: Let $\sum _{k=1}^{\infty } a_k$ and $\sum _{k=1}^{\infty } b_k$ be two series such that $0\le$_____$\le$_____ for every ______. If the series _____ converges absolutely, then the series converges absolutely.

Step-by-Step Solution

Answer

On completing the fill in the blanks, we get, "Let $\sum _{k=1}^{\infty } a_k$ and $\sum _{k=1}^{\infty } b_k$ be two series such that $0\le a_k\le b_k$ for every $k>0$. If the series $\sum _{k=1}^{\infty } b_k$ converges absolutely, then the series $\sum _{k=1}^{\infty } a_k$ converges absolutely. 

Step 1. Given ifnromation.

Consider the given question,

$\sum _{k=1}^{\infty } a_k, \sum _{k=1}^{\infty } b_k$ are the two series.

Step 2. Fill up the blanks.

Let $\sum _{k=1}^{\infty } a_k, \sum _{k=1}^{\infty } b_k$ be the two series such that $0\le a_k\le b_k$ for every $k>0$.

If the series $\sum _{k=1}^{\infty } b_k$ converges absolutely, then the series $\sum _{k=1}^{\infty } a_k$ converges absolutely.

Therefore, the first blank is completed by $a_k$, second blank by $b_k$, third blank by $k>0$, fourth blank by $\sum _{k=1}^{\infty } b_k$ and fifth by $\sum _{k=1}^{\infty } a_k$.