   Chapter 11, Problem 7RCC

Chapter
Section
Textbook Problem

# (a) If a series is convergent by the Integral Test, how do you estimate its sum? (b) If a series is convergent by the Comparison Test, how do you estimate its sum? (c) If a series is convergent by the Alternating Series Test, how do you estimate its sum?

(a)

To determine

To describe: The sum of the series when the series is convergent by the Integral Test.

Explanation

Result used: The Integral Test.

If the function f(x) is continuous, positive and decreasing on [1,) and let an=f(n) ,  then the series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

Consider the given series is convergent by the Integral Test.

That is, the series satisfies the conditions of the Integral Test.

Thus, by Result stated above, to estimate of the series sum by integrating the function from the 0 to infinity.

Example:

Consider the series n=1n3

The function from given series x3 .

The derivative of the function is obtained as follows,

f(x)=(3)x(31)=(3x4)<0

Therefore, the given function is decreasing by using the Result (2).

Clearly, the function f(x) is continuous, positive and decreasing on [1,) .

Use the Integral test the series is convergent if the improper integral 1x3dx is convergent

(b)

To determine

To describe: The sum of the series when the series is convergent by the Comparison Test.

(c)

To determine

To describe: The sum of the series when the series is convergent by the Alternating Series Test.

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