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For a convergent series satisfying the conditions of the integral test, why is every remainder Rnpositive? How can Rnbe used along with the term Sn from the sequence of partial sums to understand the quality of the approximation Sn?

Short Answer

Expert verified

The remainder is positive because the function is positive.

Then

Snk=1a(k)Sn+BnwhereBn=na(x)dx

If the remainder is small, the quality of approximation is good.

Step by step solution

01

Step 1. Given Information.

The convergent series satisfying the conditions of the integral test.

02

Step 2. Approximating the Remainder for a Series That Converges by the Integral Test.

If a function a is continuous, positive, and decreasing, and if the improper integral 1a(x)dx converges, then the nth remainder, Rn, for the series a(k)k=1is bounded by,

0Rnna(x)dx.

The remainder is positive because the function is positive.

Then

Snk=1a(k)Sn+BnwhereBn=na(x)dx

If the remainder is small, the quality of approximation is good.

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