The Unapologetic Mathematician

Mathematics for the interested outsider

Abel’s Partial Summation Formula

When we consider an infinite series we construct the sequence of partial sums of the series. This is something like the indefinite integral of the sequence of terms of the series.

What’s the analogue of differentiation? We simply take a sequence A_n and write a_0=A_0 and a_n=A_n-A_{n-1} for n\geq1. Then we can take the sequence of partial sums

\displaystyle\sum\limits_{k=0}^na_k=A_0+\sum\limits_{k=1}^n(A_k-A_{k-1})=A_n

Similarly, we can take the sequence of differences of a sequence of partial sums

\displaystyle\sum\limits_{k=0}^0a_k=a_0
\displaystyle\sum\limits_{k=0}^na_k-\sum\limits_{k=0}^{n-1}a_k=a_n

This behaves a lot like the Fundamental Theorem of Calculus, in that constructing the sequence of partial sums and constructing the sequence of differences invert each other.

Now how far can we push this analogy? Let’s take two sequences, a_n and B_n. We define the sequence of partial sums A_n=\sum_{k=0}^na_k and the sequence of differences b_0=B_0 and b_n=B_n-B_{n-1}. We calculate

\displaystyle\sum\limits_{k=0}^na_kB_k=A_0B_0+\sum\limits_{k=1}^n(A_k-A_{k-1})B_k=
\displaystyle=A_0B_0+\sum\limits_{k=1}^nA_kB_k-\sum\limits_{k=0}^{n-1}A_kB_{k+1}=
\displaystyle=\sum\limits_{k=0}^nA_kB_k-\sum\limits_{k=0}^nA_kB_{k+1}+A_nB_{n+1}=
\displaystyle=A_nB_{n+1}-\sum\limits_{k=0}^nA_k(B_{k+1}-B_k)=A_nB_{n+1}-\sum\limits_{k=0}^nA_kb_{k+1}

This is similar to the formula for integration by parts, and is referred to as Abel’s partial summation formula. In particular, it tells us that the series \sum_{k=0}^\infty a_kB_k converges if both the series \sum_{k=0}^\infty A_kb_{k+1} and the sequence A_nB_{n+1} converge.

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April 30, 2008 - Posted by | Analysis, Calculus

2 Comments »

  1. There is an interesting application of Abel’s summation to a reformulation of the Riemann Hypothesis in terms of the Mobius function.

    Comment by misha | May 1, 2008 | Reply

  2. [...] and Abel’s Tests We can now use Abel’s partial summation formula to establish a couple other convergence [...]

    Pingback by Dirichlet’s and Abel’s Tests « The Unapologetic Mathematician | May 1, 2008 | Reply


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