Chapter 8.1, Problem 107E

To determine

To write: The sum using sigma notation and to find the sum using graphing utility.

Answer to Problem 107E

The sigma notation of the given sequence is ${\displaystyle \sum _{i=1}^5\frac{2^i-1}{2^{i+1}}}$ .

The sum of the sequence using a graphing utility is $2.015625$ .

Explanation of Solution

Given information:$$

  $\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$

Calculation:

Finding the sigma notation for the given sequence,

The denominator increases exponentially by a factor $2$ which starts from $2^2=4$ ,

Therefore the denominator can be represented by $2^{i+1}$ .

The numerator increases exponentially by a factor $2$ and subtracted by $-1$ ,

Therefore the numerator can be represented by $2^i-1$ .

The sigma notation for the given sum is,

  $\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}={\displaystyle \sum _{i=1}^5\frac{2^i-1}{2^{i+1}}}$

The sum of sequence using a graphing utility is,

  $\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}={\displaystyle \sum _{i=1}^5\frac{2^i-1}{2^{i+1}}}=2.015625$

Thus we can use the sigma notation to represent a sum and find the sum using graphing utility.