The infinite spell of 2016

Algebra Level 3

Let p ( m ) = ∑ k = 0 ∞ 1 m k \displaystyle p(m) = \displaystyle \sum_{k=0}^\infty \dfrac1{m^k} and let x = ∏ r = 2 2016 p ( r ) \displaystyle x = \prod_{r=2}^{2016} p(r) . Find x ! 2014 ! ⋅ 1 2015 \dfrac{x!}{2014!} \cdot \dfrac1{2015} .


The answer is 2016.

Shivam K by Shivam K , 21, India

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2 solutions

Shivam K
Mar 14, 2016

by geometric progression with ratio <1

p(m) = m/m-1

therefore x = (2/1) * (3/2) ...... (2016/2015)

x = 2016

thus the final ans is 2016!/2015!

=2016

3 Helpful 0 Interesting 0 Brilliant 0 Confused

m is an integer here* . or else the GP is not allowed.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

m is an integer see the function in accordance with the second equation the no.s 2- 2016 are integers hence it would follow an infinite geometric progression

Shivam K - 5 years, 3 months ago

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