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Chapter 7: Problem 33

Write the series explicitly and evaluate the sum. $$ \sum_{k=0}^{3} \log \left(k^{2}+2\right) $$

Short Answer

Expert verified
The sum of the series can be written explicitly as \( \log(2) + \log(3) + \log(6) + \log(11) \), and it cannot be simplified further.

Step by step solution

01

Write the series explicitly

First, we will compute the logarithm for each value of k from 0 to 3, following the formula: \[ \log \left(k^{2}+2\right) \] For k = 0: \[ \log \left(0^{2}+2\right) = \log(2) \] For k = 1: \[ \log \left(1^{2}+2\right) = \log(3) \] For k = 2: \[ \log \left(2^{2}+2\right) = \log(6) \] For k = 3: \[ \log \left(3^{2}+2\right) = \log(11) \] Now, we can write the series explicitly: \[ \log(2) + \log(3) + \log(6) + \log(11) \]
02

Evaluate the sum

Finally, we will add the individual terms of the series to find the sum: \[ \sum_{k=0}^{3} \log \left(k^{2}+2\right) = \log(2) + \log(3) + \log(6) + \log(11) \] We cannot simplify this expression any further, so the sum of the series is: \[ \log(2) + \log(3) + \log(6) + \log(11) \]

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