The Integral Test

10.3 The Integral Test

Introduction to Series Convergence

• The main question regarding series with nonnegative terms is whether they converge.

• A series of nonnegative terms converges if the sequence of its partial sums is bounded from above.

• We examine the error involved in using partial sums to approximate the total sum of a convergent series.

Nondecreasing Partial Sums

• For an infinite series with nonnegative terms, each partial sum is at least the same as its predecessor, resulting in a nondecreasing sequence.

• Corollary of Theorem 6 (Monotonic Sequence Theorem): A series of nonnegative terms converges if and only if its partial sums are bounded from above.

Example 1: Harmonic Series

• The harmonic series diverges despite the fact that the nth term (1/n) approaches zero.

• The partial sums grow without bound, shown through grouping terms:

  • 1st group: 1, 2nd group: 1/3 + 1/4 > 1/2, 3rd: 1/5 to 1/8 > 1/2, etc.

  • General trend: The sum of 2n terms is > 1/2, making the partial sums unbounded, confirming divergence.

The Integral Test

• The Integral Test is applied to series in the context of functions.

• Example 2: Analyze convergence of ( (1/n^2) ) by comparing with the function f(x) = 1/x², interpreting values as rectangle areas under the graph.

• This shows the series converges because the sum of areas is bounded from above (by 2).

Theorem 9: The Integral Test

• For a series ( a_n = f(n) ) where f is a continuous, positive, decreasing function:

  • If the series converges, so does the integral ( , \int f(x) , dx )

  • The reverse is also true.

Example 3: The p-Series Test

• A p-series ( \sum_{n=1}^{\infty} \frac{1}{n^p} ) converges if p > 1; diverges if p ≤ 1.

  • For p > 1, ( f(x) = 1/x^p ) is positive and decreasing, confirming convergence.

  • If p = 1: Harmonic series diverges.

Example 4: Non p-Series Convergence

• For the series of ( 1/(x^2 + 1) ):

  • f(x) is positive, continuous, decreasing for x ≥ 1.

  • Converges by the Integral Test but does not specify the series' sum.

Convergence and Divergence Assessment

Example 5: Applications of Integral Test

• Example (a): Series converges; (b): Series diverges based on the Integral Test results.

Error Estimation in Convergent Series

• Notably, we cannot always determine the exact sum for a convergent series.

• We can estimate the size of the remainder ( R_n = S - S_n ) using the bounds found in the Integral Test, which helps infer the error in approximation.

Bounds for the Remainder

• For positive terms, we establish:

  • Lower and upper bounds can be set for the remainder Rn.

  • This approach assists in determining how close ( S_n ) is to ( S ), the total sum.

Example 6: Estimating the Series Sum

• Estimate the sum of the series ( (1/n^2) ) using previous results with n=10 to yield bounds: 1.64068 < S < 1.64977.

• Resulting approximation provides an error margin, emphasizing precision in summation estimation.

Conclusion and Exercises

• Utilize the Integral Test to assess convergence/divergence for diverse series.

• Verify conditions of the Integral Test are satisfied in each case.