In Oppenheim and Willsky's Systems and Signals, second edition, on page 6:

$$ E_{\infty} \stackrel{\Delta}= \lim_{t\to\infty}\int_{-T}^{T}{|x(t)|^2}dt = \int_{-\infty}^{+\infty}{|x(t)|^2} \, \mathrm{d}t \tag{Eq. 1.6}$$

$$ E_{\infty} \stackrel{\Delta}= \lim_{N\to\infty}\sum_{n=-N}^{N}{|x[n]|^2} = \sum_{-\infty}^{+\infty}{|x[n]|^2} \tag{Eq. 1.7} $$

Note: $\stackrel{\Delta}=$ means "defined as". Also, $E_\infty$ is the energy of some signal or physical quantity related to a signal.

On the following page, it says that for some signals, Eqns 1.6 and 1.7 might not converge, which confused me for Eqn 1.6. Can someone show what testing for convergence looks like for an integral? I'm assuming that we can write the integral in it's $\sum$ form and divide $\Delta t$ over and then run the various tests for divergence/convergence, but I'm not certain on that.



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