So Many Geometric Sequences

Algebra Level 3

$\begin{aligned} &\left( 1 + \dfrac12 + \dfrac14 + \dfrac18 + \cdots\right) \times\left( 1 + \dfrac13 + \dfrac19 + \dfrac1{27} + \cdots \right) \\\\ &\times \left( 1 + \dfrac14 + \dfrac1{16} + \dfrac1{64} + \cdots \right)\times \cdots \times\left( 1 + \dfrac1n + \dfrac1{n^2} + \cdots \right) \end{aligned}$

The expression above represents the product of infinite geometric progression sums .

Simplify this expression.

n^2

n

n^3

None of these options

1 solution

Chew-Seong Cheong
Nov 3, 2016

$\begin{aligned} P & = \left(1+\frac 12+ \frac 14+ \frac 18 +...\right) \left(1+\frac 13+ \frac 19+ \frac 1{27} +...\right) \left(1+\frac 14+ \frac 1{16}+ \frac 1{64} +...\right) ... \left(1+\frac 1n+ \frac 1{n^2}+ \frac 1{n^3} +...\right) \\ & = \frac 1{1-\frac 12} \times \frac 1{1-\frac 13} \times \frac 1{1-\frac 14} \times ... \times \frac 1{1-\frac 1n} \\ & = 2 \times \frac 32 \times \frac 43 \times ... \times \frac n{n-1} \\ & = \cancel{2} \times \frac {\cancel{3}}{\cancel{2}} \times \frac {\cancel{4}}{\cancel{3}} \times ... \times \frac n{\cancel{n-1}} \\ & = \boxed{n} \end{aligned}$

So Many Geometric Sequences

1 solution

0 pending reports