What is the final value of $\int_0^{\pi/2}\cos x\,dx$ in this worked example?

Partial sums approximate the function by truncating the infinite series after a finite number of terms to create a polynomial. As more terms are added, the polynomial matches the target function's graph more closely over a wider interval around the origin.

Conditions: The approximation is evaluated near the center of expansion ($x=0$).; A finite number of terms is selected from the infinite series.

The trailing ellipsis indicates that the series continues indefinitely following the exact same generating rule established by the first few terms. It means you keep substituting successive integers for $k$ into the formula $\frac{(-1)^k x^{2k}}{(2k)!}$ to produce the next terms, alternating signs and increasing the even powers and factorials.

Conditions: The first few terms have been explicitly calculated to establish a pattern.; The summation is an infinite series.

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 $(2n+1)!$ denominator comes from combining the original factorial $(2n)!$ from the cosine series coefficient with the new factor $2n+1$ introduced by the power rule of integration. Algebraically, $(2n)! \times (2n+1) = (2n+1)!$.

Conditions: The integration is performed on the term $x^{2n}$.; The original coefficient contains $(2n)!$ in the denominator.; The power rule is applied, raising the exponent to $2n+1$ and dividing by $2n+1$.

To expand the summation, substitute successive integer values for the index $k$ into the general term. For $k=0$, the term simplifies to $1$.

Conditions: The summation index $k$ starts at $0$ and increases by $1$ for each term.; The general term is $\frac{(-1)^k x^{2k}}{(2k)!}$.