Can Abel's test and Dirichlet test be used interchangably?

Question

I have seen in most cases in series of constants and in series of functions that where Abel's test of convergence applies Dirichlet's test also applies and vice versa. This makes me to think whether the two tests are equivalent in the sense that one is derivable from other.Is it true? Or there are cases where we can apply one but cannot apply other. Also is it so that one is more general and other is a corollary out of it.

Answer 1

Neither test is more general than the other, each extends their common domain in a different direction than the other. Thus there are situations where Abel's test is applicable but Dirichlet's isn't, and there are situations where Dirichlet's test is applicable but not Abels's test.

Abel's test:

If $(a_n)$ and $(b_n)$ are sequences of real-valued functions on a set $X$, such that the series $\sum a_n(x)$ converges uniformly, and $(b_n)$ converges uniformly and pointwise monotonically (so $(b_n(x))$ is monotonic for each $x$, but it may be nonincreasing for some $x$ and nondecreasing for others) to the bounded function $b$, then $\sum a_n(x)b_n(x)$ converges uniformly.

Dirichlet's test:

If $(a_n)$ and $(b_n)$ are sequences of real-valued functions on a set $X$, such that the series $\sum a_n(x)$ has uniformly bounded partial sums, and $(b_n)$ converges uniformly and pointwise monotonically to $0$, then $\sum a_n(x)b_n(x)$ converges uniformly.

So it's easy to see that Abel's test isn't applicable for $a_n = (-1)^n$ and $b_n = \frac{1}{n}$ while Dirichlet's test is, and Abel's test is applicable for $a_n = \frac{(-1)^n}{n}$ and $b_n = 1 + 2^{-n}$ while Dirichlet's isn't.

However, it is straighforward to deduce Abel's test from Dirichlet's:

Let $\beta_n = b_n - b$. Then $$\sum a_n(x)b_n(x) = \sum a_n(x)b(x) + \sum a_n(x)\beta_n(x)$$ converges uniformly because each of the two series on the right does. The first by the assumption on $(a_n)$ and the boundedness of $b$, the second by Dirichlet's test.

In that sense, Abel's test is a corollary of Dirichlet's.

I don't see a way to deduce Dirichlet's test from Abel's without essentially proving Dirichlet's test from scratch. (Of course that doesn't prove there isn't.)

Answer 2

The proofs have some commonality but one test is not a corollary of the other.

Suppose the sequence $(a_n)$ is not monotone and $\sum a_n$ converges. It follows that $\sum(1+1/n)^n a_n$ converges by Abel’s test. However, the conditions for Dirichlet’s test are not met since $(1+1/n)^n$ is not monotone decreasing — although it is bounded and monotone increasing.