Calculating the Value of $f(m,n)$

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Discussion Overview

The discussion centers around the function $f(m,n) = 3m+n+(m+n)^2$ and the evaluation of the double sum $\displaystyle \sum_{n=0}^{\infty}\;\; \sum_{m=0}^{\infty}2^{-f(m,n)}$. Participants explore the behavior of this function and its implications for the convergence of the sum.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant suggests evaluating the sum numerically, noting that it converges quickly and approximates to about 1.33333 with just four terms of each summation.
  • Another participant emphasizes the importance of examining small values of $m$ and $n$, proposing to create a table of values for $f(m,n)$ to uncover patterns.
  • A participant questions the origin of the problem, revealing it is from a Putnam competition, although the specific year is not recalled.
  • There is a mention of uncertainty regarding a previous calculation that yielded an incorrect answer.

Areas of Agreement / Disagreement

Participants express varying approaches to the problem, with no consensus on a definitive method or solution. The discussion remains exploratory with multiple viewpoints on how to tackle the evaluation of the sum.

Contextual Notes

Participants have not resolved the implications of the function's behavior or the correctness of the numerical approximation. There are also unresolved questions regarding the previous calculations mentioned.

juantheron
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Let $f(m,n) = 3m+n+(m+n)^2.$ Then value of $\displaystyle \sum_{n=0}^{\infty}\;\; \sum_{m=0}^{\infty}2^{-f(m,n)}=$
 
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Take a look at $f(n,m).$
 
jacks said:
Let $f(m,n) = 3m+n+(m+n)^2.$ Then value of $\displaystyle \sum_{n=0}^{\infty}\;\; \sum_{m=0}^{\infty}2^{-f(m,n)}=$

Might I ask where this question comes from?

The sum converges very quickly and can be evaluated numerically with 4 terms of each summation (it is \(\approx 1.33333\), which is very suggestive ... ) to good accuracy.

CB

(why the previous calc got the wrong answer I still have no idea)
 
Last edited:
I hope Krizalid will not object if I add a bit to his very helpful suggestion. In problems like this, my advice is always to start by looking at what happens for small values of the variables. In this case, if you make a table of the values of $f(m,n)$ for small values of $m$ and $n$, it looks like this:

$$\begin{array}{cc}&\;\;\;\;n \\ \rlap{m} & \begin{array}{c|cccc} &0&1&2&3 \\ \hline 0&0&2&6&12 \\ 1&4&8&14&. \\ 2&10&16&.&. \\ 3&18&.&.&. \end{array} \end{array}$$

Doesn't that suggest something very interesting about the range of the function $f(m,n)$?
 
CaptainBlack said:
Might I ask where this question comes from?
It's from a Putnam. I don't remember the year though.
 

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