$\"\"$

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(a).

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The series is $\"\"$ .

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Find the sum of the first ten terms of the series.

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The sum of the first ten terms is $\"\"$ .

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$\"\"$

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Estimate the error using Remainder estimate for the integral test.

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$\"\"$

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substitute $\"\"$ .

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The error is $\"\"$ which is not greater than $\"\"$ .

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$\"\"$

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The error is not more than $\"\"$ .

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$\"\"$

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(b).

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Equation $\"\"$ : $\"\"$ .

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Here $\"\"$ , since $\"\"$ and $\"\"$ .

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Substiute the values.

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$\"\"$

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Consider $\"\"$ .

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ .

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Therefore $\"\"$

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$\"\"$ .

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Consider $\"\"$ as the average of the upper and the lower bounds , we have

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$\"\"$ and with error

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$\"\"$ .

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$\"\"$

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(c).

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The value of $\"\"$ and error $\"\"$ .

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From the result of exercise 35, $\"\"$ .

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Difference with Eulers exact value : $\"\"$ .

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$\"\"$

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The value is within the range of error calulated for the estimate.

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$\"\"$

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(d).

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The function $\"\"$ , since $\"\"$ .

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Estimate the error using Remainder estimate for the integral test.

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$\"\"$ the error is $\"\"$ is no more than $\"\"$ .

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Therefore $\"\"$ , we need to find the value of $\"\"$ so that $\"\"$ .

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$\"\"$

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The value of $\"\"$ .

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$\"\"$

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$\"\"$

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Since $\"\"$ has to be an integer , so $\"\"$ is atleast $\"\"$ , i.e., $\"\"$ .

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$\"\"$

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(a)The error is not more than $\"\"$ .

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(b) error $\"\"$ .

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(c) $\"\"$ . \ \

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(d) $\"\"$ . $\"\"$