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Chapter 14: Problem 37

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{14}{14+1}$$

Short Answer

Expert verified
The given sum can be expressed using summation notation as \( \sum_{i=1}^{14}\frac{i}{i+1} \)

Step by step solution

01

Identify the pattern

Looking at the provided sequence, there appears to be a clear pattern. The numerator for each fraction is an integer that increases by one for each subsequent term. The corresponding numerator plus one makes up the denominator. The sequence starts at \(\frac{1}{2}\) and ends at \(\frac{14}{15}\) with each term being the addition of the previous fraction and the next fraction following the pattern.
02

Write the pattern as a function of i

The pattern can be expressed as a function of i as \( \frac{i}{i+1} \) for every i from 1 to 14.
03

Express in summation notation

The summation from i = 1 to 14 inclusive for \( \frac{i}{i+1} \) can be expressed in summation notation as \( \sum_{i=1}^{14}\frac{i}{i+1} \)

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(10-5+\frac{5}{2}-\frac{5}{4}+\cdots=\frac{10}{1-\frac{1}{2}}\)

Use the formula for the sum of the first n terms of a geometric sequence to solve. You save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, continuing to double your savings each day. What will your total savings be for the first 30 days?

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