**Mathematical Convergence Inquiry**This question explores a concept in mathematical series convergence:"If $ \sum a_N $ with $ a_N > 0 $ is convergent, then is $ \sum \sqrt{a_N a_{N+1}} $ always convergent?"**Explanation:**The expression begins by stating a condition for convergence. It examines whether the convergence of the series $ \sum a_N $, with each term $ a_N $ being positive, implies the convergence of another related series $ \sum \sqrt{a_N a_{N+1}} $.**Understanding the Series:**- **$ \sum a_N $:** This is a traditional series of terms $ a_N $ that converges (sums to a finite value) with each $ a_N $ being greater than zero.- **$ \sum \sqrt{a_N a_{N+1}} $:** This is a new series created by taking the square root of the product of consecutive terms from the original series. The problem asks if this series also converges under the given condition.The problem invites exploration into the dependencies and conditions affecting the convergence of transformed series, fostering a deeper understanding of series behavior.

Given, is convergent, where, So, is.Now, using A.M≥G.M property,We have,

Answered: If an with AN>O is Convergent, then is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Step 1: Explanation

Given, $sum from n to blank of a subscript n$ is convergent, where, $a subscript n greater than 0$

So, $sum for n of a subscript n plus 1 end subscript$ is.

Now, using A.M≥G.M property,

We have,

$fraction numerator a subscript n plus 1 end subscript plus a subscript n over denominator 2 end fraction greater or equal than square root of a subscript n plus 1 end subscript a subscript n end root$

$rightwards double arrow sum from n to blank of square root of a subscript n plus 1 end subscript a subscript n end root less or equal than sum from n to blank of fraction numerator a subscript n plus a subscript n plus 1 end subscript over denominator 2 end fraction$

$rightwards double arrow sum for n of square root of a subscript n plus 1 end subscript a subscript n end root less or equal than 1 half sum for n of a subscript n plus 1 half sum for n of a subscript n plus 1 end subscript$