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The sum and integral can be interchanged because the Dominated Convergence Theorem applies. The terms of the cosine power series are bounded in magnitude by $1$ on the interval $[0,\pi/2]$, making the constant function $g(x)=1$ a valid dominating function with a finite integral over the interval.

Conditions: The integration interval is $[0,\pi/2]$.; The series terms are $\frac{(-1)^n}{(2n)!}x^{2n}$.; The dominating function is $g(x)=1$.

The sum and integral can be interchanged because the Dominated Convergence Theorem applies. The terms of the cosine power series are bounded in magnitude by $1$ on the interval $[0,\pi/2]$, making the constant function $g(x)=1$ a valid dominating function with a finite integral over the interval.

Conditions: The integration interval is $[0,\pi/2]$.; The series terms are $\frac{(-1)^n}{(2n)!}x^{2n}$.; The dominating function is $g(x)=1$.

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.