  The diametral dimension, $\Delta(E)$, and the approximate diametral
dimension, $\delta (E)$, of a nuclear Fr\'echet space $E$ which satisfies
$\underline{DN}$ and $\Omega$, is related to power series spaces
$\Lambda_{1}(\varepsilon)$ and $\Lambda_{\infty}\left(\varepsilon\right)$ for
some exponent sequence $\varepsilon$. In this article, we examine a question of
whether $\delta (E)$ must coincide with that of a power series space if
$\Delta(E)$ does the same, and vice versa. In this regard, we first show that
this question has an affirmative answer in an infinite type case by proving the
fact that $\Delta (E)=\Delta\left(\Lambda_{\infty} (\varepsilon)\right)$ if and
only if $\delta (E)= \delta (\Lambda_{\infty}(\varepsilon))$. Then we consider
the question in the finite type case and, among other things, we prove that
$\delta (E)=\delta\left(\Lambda_{1} (\varepsilon)\right)$ if and only if
$\Delta (E)= \Delta (\Lambda_{1}(\varepsilon))$ and $E$ has a prominent bounded
subset.

Functional Analysis