An easy nut to crack! Right?

Calculus Level 4

∑ n = 1 ∞ n 4 4 n = p q \displaystyle \sum_{n=1}^{\infty}\frac{n^{4}}{4^{n}}=\frac{p}{q}

If p p and q q are co-prime positive integers. Find p + q p+q .


The answer is 461.

Akshat Sharda by Akshat Sharda , 21, India

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2 solutions

It was a lengthy one for me as I don't know Polylogarithm. I used the common method of dividing the summision and subtracting it from the pre-summision, until you don't get a constant Geometric series.

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Isaac Buckley
Aug 31, 2015

For these types of problems we just use the Polylogarithm :

Li − 4 ( 1 4 ) = 380 81 \LARGE \text{Li}_{-4}\left(\frac1{4}\right)=\frac{380}{81}

2 Helpful 0 Interesting 0 Brilliant 1 Confused

Hahahahaha! What an answer!

Pi Han Goh - 5 years, 6 months ago

And how do you further evaluate the resulting poly logarithm?

A Former Brilliant Member - 4 years, 8 months ago

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See this , under Li ( − 4 ) ( z ) \text{Li}(-4) (z) .

Pi Han Goh - 4 years, 8 months ago

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