summation by part and dirichlet test

hantian

June 27, 2024

it is often-pactised to introduce integral by parts via the product rule. intuitively

d uv = u d v + v d u,

this is a perfect starting point. however students tends to overlook the other side of this result, that is, the implication of summation.

indeed, the theorem in concern can be proven in a pure integral form:

\int u v^{'} d x = \int u d v

discretise the above,

\begin{align} \sum_{i = 1}^{n} u_{i} (v_{i + 1} - v_{i}) & = \sum_{i = 1}^{n} u_{i} v_{i + 1} - \sum_{i = 1}^{n} u_{i} v_{i} \\ = \sum_{i = 2}^{n} v_{i} (u_{i - 1} - u_{i}) - v_{1} u_{1} + v_{n + 1} u_{n} . & (1) \end{align}

write (1) back into integral form, that is

uv - \int v d u = uv - \int u^{'} v d x .

1 dirichlet test of convergence

equation (1) by itself is interesting, as it offers an alternative way to calculate infinite series, this was first introduced by abel¹ as a discrete analog to integral by parts.

however, an important question that pursuing the formula is the convergence of the summation itself. now with abel’s summation, comes the following dirichlet test²

the summation $\sum a_{n} b_{n}$ converges if series $b_{n}$ is absolute convergent $\sum | b_{n} | < M$ and series $a_{n}$ decreases to zero $\lim a_{n} = 0$ .

the result is only natural with the abel’s summation. consider the partial sum

\begin{array}{l} S_{N} & = \sum_{i = 1}^{N} a_{i} b_{i} \\ = \sum_{i = 1}^{N} a_{i} (B_{i} - B_{i - 1}) \\ = a_{n} B_{n} - a_{1} B_{1} - \sum_{i = 2}^{n - 1} B_{i} (a_{i} - a_{i - 1}) . \end{array}

obviously, if the conditions are satisfied

\lim_{N \to \infty} | S_{N} | < \lim_{N \to \infty} a_{N} B_{N} - a_{1} B_{1} + M (a_{1} - a_{N - 1}) = | M - b_{1} | a_{1} .

it is also possible to extend this test to continuous functions, as to test the convergence of improper integrals.

integral $\int fg d x$ converges if $\int f d x$ is uniformly bounded and none-negative function $g$ monotonically decreasing.

both dirichlet’s test and abel’s test which is delibrately ommitted in this post have important significance in real analysis, that is, when $a_{n}$ and $b_{n}$ are functions of $x$ .