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Chapter 7: Problem 32

Write the series explicitly and evaluate the sum. $$ \sum_{m=1}^{5}\left(m^{2}-2 m+7\right) $$

Short Answer

Expert verified
The explicit series is given by \(6 + 7 + 10 + 15 + 22\). The sum of the series \(\sum_{m=1}^{5}\left(m^{2}-2 m+7\right)\) is \(60\).

Step by step solution

01

Write the series explicitly

To write the series explicitly, substitute the range of values for m (from 1 to 5) into the given expression \(m^2 - 2m + 7\), and write the terms as follows: For m = 1: \(1^2 - 2(1) + 7 = 6\) For m = 2: \(2^2 - 2(2) + 7 = 7\) For m = 3: \(3^2 - 2(3) + 7 = 10\) For m = 4: \(4^2 - 2(4) + 7 = 15\) For m = 5: \(5^2 - 2(5) + 7 = 22\) Now, we can rewrite the series as: \[ \sum_{m=1}^{5}\left(m^{2}-2 m+7\right) = 6 + 7 + 10 + 15 + 22 \]
02

Evaluate the sum of the series

Now that the series is written explicitly, we can simply add up all the terms to find the sum: \[ 6 + 7 + 10 + 15 + 22 = 60 \] So, the sum of the series \(\sum_{m=1}^{5}\left(m^{2}-2 m+7\right)\) is \(60\).

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