(a).

The series is .

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

The sum of the first ten terms is .

Estimate the error using Remainder estimate for the integral test.

substitute .

The error is which is not greater than .

The error is not more than .

(b)

Equation : .

Here , since and .

Substiute the values.

Substitute and .

Therefore

.

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

and with error

.

(c).

The value of and error .

Difference with Eulers exact value :.

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

(d).

The function , since .

Estimate the error using Remainder estimate for the integral test.

the error is   is no more than  .

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

The value of .

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

(a).The error is not more than .

(b). error .

(c).  .

(d). .