Abel Summation by Parts

Worked Example

Consider the sequence a_k = (-1)^(k-1) and weights b_k = 1/k. We compute the weighted sum for N = 4 terms:

a = [1, -1, 1, -1]
b = [1, 1/2, 1/3, 1/4]

Weighted sum = (1)(1) + (-1)(1/2) + (1)(1/3) + (-1)(1/4)
             = 1 - 1/2 + 1/3 - 1/4
             = 12/12 - 6/12 + 4/12 - 3/12
             = 7/12 ≈ 0.5833333333333334

Partial sums A_k = [1, 0, 1, 0]

Boundary term = A_4 * b_4 = (0)(1/4) = 0

Correction sum = sum_{k=1}^3 A_k * (b_k - b_{k+1})
               = A_1(b_1 - b_2) + A_2(b_2 - b_3) + A_3(b_3 - b_4)
               = 1(1 - 1/2) + 0(1/2 - 1/3) + 1(1/3 - 1/4)
               = 1(1/2) + 0 + 1(1/12)
               = 1/2 + 1/12
               = 6/12 + 1/12
               = 7/12 ≈ 0.5833333333333334

The boundary term plus the correction sum equals the weighted sum, verifying the identity.

Experiment

In the Abel Summation by Parts module, you can explore the relationship between the weighted sum, boundary term, and correction sum for different sequences and weights.

To see this in action:

1. Set the Sequence control to Alternating.
2. Set the Weights control to Reciprocal.
3. Observe the Partial sums plot and note that they alternate between 1 and 0, ensuring they are bounded.
4. Increase the Terms control and observe the Boundary term output approach 0, illustrating Dirichlet's test for convergence.

Limitations

Dirichlet convergence is only guaranteed when the partial sums are bounded and the weights decrease monotonically to 0. Not every selectable sequence and weight combination meets these hypotheses. For example, using Constant sequence and Linear weights results in an unbounded weighted sum. However, the finite algebraic identity always holds exactly, regardless of convergence properties.

References

Further reading on summation by parts and related tests for convergence:

• NIST Digital Library of Mathematical Functions: Section 2.10