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The integrated cosine series becomes the sine series because integrating the even-power terms $x^{2n}$ of the cosine series produces odd-power terms $x^{2n+1}$. Combined with the factorial simplification $(2n)!(2n+1) = (2n+1)!$, the resulting series exactly matches the standard Maclaurin series for $\sin x$.

Conditions: The starting series is the Maclaurin series for $\cos x$.; Term-by-term integration is justified by the DCT.; The factorial identity $(2n)!(2n+1) = (2n+1)!$ is applied.

The integrated cosine series becomes the sine series because integrating the even-power terms $x^{2n}$ of the cosine series produces odd-power terms $x^{2n+1}$. Combined with the factorial simplification $(2n)!(2n+1) = (2n+1)!$, the resulting series exactly matches the standard Maclaurin series for $\sin x$.

Conditions: The starting series is the Maclaurin series for $\cos x$.; Term-by-term integration is justified by the DCT.; The factorial identity $(2n)!(2n+1) = (2n+1)!$ is applied.

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 $g(x)=1$.