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Chapter 14: Problem 39

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

Short Answer

Expert verified
The given sum can be expressed in sigma notation as \( \sum_{i=1}^{n} 4^{i}/i \)

Step by step solution

01

Identify the pattern

The first thing that should be noted is the pattern in the given sum. In the given sum, each term can be written in the form of \(4^{i}/i\), where 'i' is the term's place in the sequence, starting from 1. The first term is 4, which is equal to \(4^{1}/1\). The next is \(4^{2}/2\), then \(4^{3}/3\), and so on. This pattern continues up to \(4^{n}/n\). So, the pattern is '4' raised to the index of the term divided by the index itself.
02

Write in summation notation

Having identified the pattern, the sum can be expressed using sigma notation. '1' is the lower limit as the sequence starts from 1. 'i' is the index of summation which varies from 1 to 'n'. 'n' is the upper limit of the summation. Each term in the series '4 + \(4^{2}/2\)+ \(4^{3}/3\)+...+ \(4^{n}/n\)' can be represented as \(4^{i}/i\). Thus, the given sum in summation notation is \( \sum_{i=1}^{n} 4^{i}/i \). This directs you to add up the terms generated by taking the value of 'i' starting from 1 and going up to (and including) 'n' according to the rule we found (\(4^{i}/i\)).
03

Confirm the solution

Re-substitute '1', '2', and '3' into \( \sum_{i=1}^{n} 4^{i}/i \) to verify the equivalence. The resulting sum should match the given sum, confirming the validity of the solution.

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