What is the maximum of a series with positive numbers and a constraint?

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Discussion Overview

The discussion revolves around finding the maximum of a series involving positive numbers under a specific constraint. The context includes multivariate calculus concepts, particularly focusing on optimization techniques and constraints.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant presents a problem involving the maximization of a series of positive numbers subject to a constraint on their squared sum.
  • Another participant inquires about the use of Lagrange multipliers as a potential method for solving the problem.
  • A participant acknowledges familiarity with Lagrange multipliers but expresses uncertainty about how to apply them to the given series and whether the summations can be treated as functions.
  • There is a request for clarification on how to apply the concept of Lagrange multipliers to the specific series presented.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the application of Lagrange multipliers to the problem, and uncertainty remains regarding the formulation of the summations as functions.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the functions involved and the specific application of optimization techniques to the problem.

Metahominid
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So I ran into this on my multivariate calculus final this morning, I don't think anyone did it but it has been stuck in my head and I have run out of thoughts.

Let a1, a2, …, an be n positive numbers.
Find the maximum of [tex]\sum^{n}_{i=1}a_{i}x^_{i}[/tex]subject to the constraint [tex]\sum^{n}_{i=1}x^{2}_{i} = 1[/tex]

Alright I've never used Latex but I believe that worked. Not quite sure, where to go looks like power sums of power 1,2.
 
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Yes, we studied it. About an hour ago I saw the word constraint and it popped into my head but I got discouraged when I tried to formulate it in the terms of functions. I wasn't entirely sure I could say that the summations were functions. I figure I need walk around for a moment and I will see it.
 
How do you apply that to the above series though?
 

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