10 is a big figure already

Calculus Level 4

What is the value of ∑ n = 1 ∞ n 10 2 n \displaystyle \sum _{n=1}^{\infty} \frac{n^{10}}{2^{n}} ?


The answer is 204495126.

Joel Yip by Joel Yip , 21, Singapore

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2 solutions

Akshay Yadav
Sep 26, 2015

This can be easily done using poylogarithms,

p o l y l o g [ − 10 , 2 ] = − 204495126 polylog[-10,2] = -204495126

But since the sum can never be negative,

Hence, 204495126 \boxed{204495126}

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Incredible Mind
Feb 26, 2015

this is too long and i believe shorter method exists.however this is what i did

x/(1-x) = x+x^2+x^3....

do this 10 times

diff. w.r.t x then multiply both sides by x.keep doing this and then substitute x=0.5

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Polylogarithm(-10,1/2)

Kartik Sharma - 6 years, 3 months ago

Well, don't say you actually did that!

Kartik Sharma - 6 years, 3 months ago

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