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The example is included to demonstrate that the domination hypothesis alone is insufficient if the dominating function's integral diverges. It serves as a warning that if the finite-integral condition is not met, the desired interchange of integral and sum may fail.

Conditions: The example is presented as a warning illustration on the interval $[0,\infty)$.; The function $e^{2x}$ dominates $e^x$, but $\int_0^{\infty} e^{2x}\,dx$ diverges.

The example is included to demonstrate that the domination hypothesis alone is insufficient if the dominating function's integral diverges. It serves as a warning that if the finite-integral condition is not met, the desired interchange of integral and sum may fail.

Conditions: The example is presented as a warning illustration on the interval $[0,\infty)$.; The function $e^{2x}$ dominates $e^x$, but $\int_0^{\infty} e^{2x}\,dx$ diverges.

The three hypotheses are: (1) a sequence of functions $f_n$ converges to a function $f$ on the interval $[a,b]$; (2) there exists a dominating function $g$ such that $|g(x)| \ge |f_n(x)|$ for all $n$ and all $x \in [a,b]$; and (3) the integral of the dominating function over the interval is finite, meaning $\int_a^b g(x)\,dx < \infty$.

Conditions: The theorem is presented in a calculus-oriented form rather than a full measure-theoretic proof.; The interval $[a,b]$ is a closed, bounded real interval.

The three hypotheses are: (1) a sequence of functions $f_n$ converges to a function $f$ on the interval $[a,b]$; (2) there exists a dominating function $g$ such that $|g(x)| \ge |f_n(x)|$ for all $n$ and all $x \in [a,b]$; and (3) the integral of the dominating function over the interval is finite, meaning $\int_a^b g(x)\,dx < \infty$.

Conditions: The theorem is presented in a calculus-oriented form rather than a full measure-theoretic proof.; The interval $[a,b]$ is a closed, bounded real interval.