<s>[INST] Question on Series Transformations from "The Harmonic Series and Logarithmic Intragrals" This is from an example in Section 1.7 of the book, "The Harmonic Series and Logarithmic Integrals" by, Ali Shadhar Olaikhan
$$\sum_{n=0}^\infty \frac{-1^n}{(2n+1)^3} = \mathcal{J}\sum_{n=1}^\infty \frac{i^n}{n^3}$$
I don't understand how the left side of this example has any Complex terms.  It appears to be entirely Real.  What am I not understanding, please?
 [/INST] \begin{align*}
\Im \sum_{n=1}^\infty \frac{\mathrm{i}^n}{n^3}
&= \Im \left( \frac{\mathrm{i}^1}{1^3} + \frac{\mathrm{i}^2}{2^3} + \frac{\mathrm{i}^3}{3^3} + \frac{\mathrm{i}^4}{4^3} + \cdots \right)  \\
&= \Im \frac{\mathrm{i}^1}{1^3} + \Im \frac{\mathrm{i}^2}{2^3} + \Im \frac{\mathrm{i}^3}{3^3} + \Im \frac{\mathrm{i}^4}{4^3} + \cdots   \\
&= \frac{1}{1^3} + 0 + \frac{-1}{3^3} + 0 + \cdots   \\
\end{align*}
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