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\(\displaystyle \sum_{r=1}^{\infty} \frac{6^r}{(3^r - 2^r)(3^{r + 1} - 2^{r + 1})} \)

Note by Kushagraa Aggarwal
4 years, 1 month ago

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Let's try to generalize. For \(|x| < 1\) we have the following:

\(S(x) := \displaystyle \sum_{n = 1}^{\infty} \dfrac{x^n}{(1-x^n)(1-x^{n+1})}\) \(= \dfrac{1}{1-x}\displaystyle \sum_{n = 1}^{\infty} \dfrac{x^n(1-x)}{(1-x^n)(1-x^{n+1})}\)

\(= \dfrac{1}{1-x}\displaystyle \sum_{n = 1}^{\infty} \dfrac{x^n - x^{n+1}}{(1-x^n)(1-x^{n+1})}\) \(= \dfrac{1}{1-x}\displaystyle \sum_{n = 1}^{\infty} \dfrac{(1- x^{n+1}) - (1-x^n)}{(1-x^n)(1-x^{n+1})}\)

\(= \dfrac{1}{1-x}\displaystyle \sum_{n = 1}^{\infty}\left[\dfrac{1}{1-x^n} - \dfrac{1}{1-x^{n+1}}\right]\) \(= \dfrac{1}{1-x}\left[ \dfrac{1}{1-x^1} - \dfrac{1}{1-0}\right]\) \(= \dfrac{x}{(1-x)^2}\).

Now to tackle the sum in question:

\(\displaystyle\sum_{n = 1}^{\infty} \dfrac{6^n}{(3^n-2^n)(3^{n+1}-2^{n+1})}\) \(= \displaystyle\sum_{n = 1}^{\infty} \dfrac{3^{2n}\left(2/3\right)^n}{3^{2n+1}\left(1-\left(2/3\right)^n\right)\left(1-\left(2/3\right)^{n+1}\right)}\)

\(= \dfrac{1}{3}\displaystyle\sum_{n = 1}^{\infty} \dfrac{\left(2/3\right)^n}{\left(1-\left(2/3\right)^n\right)\left(1-\left(2/3\right)^{n+1}\right)}\) \(= \dfrac{1}{3}S\left(\dfrac{2}{3}\right)\) \(= \dfrac{1}{3} \dfrac{2/3}{\left(1-2/3\right)^2} = \boxed{2}\)

Jimmy Kariznov - 4 years, 1 month ago

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The best part of this solution is that it's deceptively straight forward. If somebody were to see all of these "fancy" math symbols and such they would instinctively feel intimidated, but if you really look at what you're doing here it's really just very smart equation manipulation.

Michael Tong - 4 years, 1 month ago

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Wow! Just great, Jimmy!

How did you even think of S(x)?

Pranav Arora - 4 years, 1 month ago

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When solving this question, I first did the manipulations to the summation. I then replaced \(2/3\) with \(x\) to see if there was some clever power series manipulations that could be done. As it turned out, it was just a telescoping sum, and replacing \(2/3\) with \(x\) was not really necessary. However, \(x\) is easier to type repeatedly than \(2/3\), and so, I showed the generalization.

Jimmy Kariznov - 4 years, 1 month ago

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@Jimmy Kariznov That makes sense, thank you.

Pranav Arora - 4 years, 1 month ago

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that is incredible genius imo, you should be proud xD

Jord W - 4 years ago

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Forgive my ignorance but what is S(x)? Is it the same as f(x)?

Alex Benfield - 4 years, 1 month ago

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\(S(x)\) is a function of \(x\) that I defined. I could have also named it \(f(x)\), or \(\xi(x)\), or even \(\clubsuit(x)\).

Jimmy Kariznov - 4 years, 1 month ago

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How will we get

\( \sum_{n=1}^{\infty} \Bigg[ -\frac{1}{1-x^{n+1}} \Bigg] = -\frac{1}{1-0} \) ?

Please help , Thank You

Priyansh Sangule - 4 years, 1 month ago

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Look at the whole expression. It states

\( \sum_{n = 1}^{\infty} \frac {1}{1 - x^n} - \frac{1}{1 - x^{n + 1}}\). So, when \(n = 1\) then you have \( \frac {1}{1 - x} - \frac {1}{1 - x^2}\). Add \(n = 2\) you have \( \frac {1}{1 - x^2} - \frac {1}{1 - x^3}\). Notice how the first term of this cancels with the second term of the first. This is called a telescopic series. Thus, you will only have the first term ( \( \frac{1}{1 - x}\)) and the last term, \( \lim_{n \rightarrow \infty} - \frac{1}{1 - x^{n + 1}} = - \frac {1}{1 - 0} \).

John Cain - 4 years, 1 month ago

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@John Cain Thank you very much now I get it !

Good explanation too !

Thank You !

Priyansh Sangule - 4 years, 1 month ago

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If I'm not mistaken, this was a Putnam problem!

Jess Smith - 4 years, 1 month ago

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