   Chapter 11.3, Problem 37E

Chapter
Section
Textbook Problem

# (a) Use the sum of the first 10 terms to estimate the sum of the series ∑ n = 1 ∞ 1 / n 2 . How good is this estimate? (b) Improve this estimate using (3) with n = 10. (c) Compare your estimate in part (b) with the exact value given in Exercise 34. (d) Find a value of n that will ensure that the error in the approximation s ≈ sn is less than 0.001.

(a)

To determine

To estimate: The sum of the series by using the sum of the first 10 terms.

Explanation

Given:

The series is n=11n2 .

Result used:

(1) If the function f(k)=ak , where f is a continuous, positive and decreasing function for xn and an is convergent. If Rn=ssn , then n+1f(x)dxRnnf(x)dx

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

(3) The function is decreasing if f(x)<0 .

(4) The P series n=11np is convergent if p>1 .

Calculation:

Compute the partial sum s10 from the given series.

s10=n=1101n2=112+122+132+...+1102=1+14+...+11001.549768

Thus, the sum of the series by suing this partial sum of series is approximately 1.549768.

Consider the function from the given series as 1x2 .

Since the function is monotonic, it is continuous. Thus, the derivative of the function is obtained as follows

(b)

To determine

To improve: The estimation using (3) with n=10 of the series.

(c)

To determine

To compare: The estimated value obtained in part (b) with the exact value given in Exercise 34.

(d)

To determine

To find: The value of n so that the error in the approximation is less than 0.001.

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