Chapter 10, Problem 69RE

To determine

To prove: Using Nicole Oresme’s Theorem prove $1+\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 3+\mathrm{...}+\frac{n}{2^{n-1}}+\mathrm{...}=4$

Explanation of Solution

Given:

The given series is,

  $1+\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 3+\mathrm{...}+\frac{n}{2^{n-1}}+\mathrm{...}=4$

As the expansion of the function $\frac{1}{1-x}$ is

  $\frac{1}{1-x}=1+x+x^2+\mathrm{...}+x^n+\mathrm{...}$

On differentiating both sides with respect tox

  $\frac{d}{dx}\left(\frac{1}{1-x}\right)=\frac{d}{dx}\left(1+x+x^2+\mathrm{...}+x^n+\mathrm{...}\right)$

  $\Rightarrow \frac{1}{{\left(1-x\right)}^2}=1+2x+3x^2+\mathrm{...}+n{x}^{n-1}+\mathrm{...}$

Substitute $x=\frac{1}{2}$ in the above expression results

  $\frac{1}{{\left(1-\frac{1}{2}\right)}^2}=1+2\cdot \frac{1}{2}+3\cdot {\left(\frac{1}{2}\right)}^2+\mathrm{...}+n{\left(\frac{1}{2}\right)}^{n-1}+\mathrm{...}$

  $\Rightarrow 4=1+\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 3+\mathrm{...}+\frac{n}{2^{n-1}}+\mathrm{...}$

Hence, the required expression is proved.