Chapter 4, Problem 10C

Explanation of Solution

To prove:

${\displaystyle \sum _{i=1}^n\frac{i}{2^i}}<2$

Proof:

Let us prove the above case by induction steps.

Base case: Let us consider the value of “n” is “0”. Substituting the value of “n” in the integral results:

$S={\displaystyle \sum _{i=1}^0\frac{1}{2^1}}<2$

Thus, from the above integral the value of “S” is less than 2.

Induction case:

Taking the summation for following case:

$S=\left({\displaystyle \sum _{i=1}^n\frac{1}{2^i}}\right)+({\displaystyle \sum _{i=2}^n\frac{\left(i-1\right)}{2^i}}$