Where does the $(2n+1)!$ denominator come from after integrating $x^{2n}$ ?

Q: Where does the $(2n+1)!$ denominator come from after integrating $x^{2n}$?

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, $(2n)! \times (2n+1) = (2n+1)!$.

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 $2n+1$ .

Reasoning, step by step

Start with the integrated term: $\frac{(-1)^n}{(2n)!} \cdot \frac{x^{2n+1}}{2n+1}$ .
Group the denominator factors: $(2n)! \times (2n+1)$ .
Apply the factorial recurrence relation: $(2n)! \times (2n+1) = (2n+1)!$ .
Rewrite the term as $\frac{(-1)^n}{(2n+1)!} x^{2n+1}$ .
Recognize this as the general term of the sine power series.

Example

The speaker says the denominator “two n factorial times two n plus one” can be pulled together into one factorial, “two n plus one factorial.”

Common misconceptions

Forgetting that the power rule introduces a division by the new exponent, which must be multiplied with the existing factorial.
Assuming that $(2n)! + (2n+1) = (2n+1)!$ , confusing addition with multiplication in the factorial recurrence.

Watch the explanation

YouTubeDominated Convergence Theorem

2:57 – 3:09Watch this moment ↗

Connected concepts

Power series →

Where does the $(2n+1)!$ denominator come from after integrating $x^{2n}$ ?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

Related questions

Where does the (2n+1)!(2n+1)!(2n+1)! denominator come from after integrating x2nx^{2n}x2n?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

Related questions

How do partial sums approximate the function defined by the series 1−x22!+x44!−x66!+…1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots1−2!x2​+4!x4​−6!x6​+…?+

What does the trailing ellipsis mean in the expanded series 1−x22!+x44!−x66!+…1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots1−2!x2​+4!x4​−6!x6​+…?+

What is the final value of ∫0π/2cos⁡x dx\int_0^{\pi/2}\cos x\,dx∫0π/2​cosxdx in this worked example?+

Why does the integrated cosine series become the sine series?+

How do you expand the summation ∑k=0∞(−1)kx2k(2k)!\sum_{k=0}^{\infty}\frac{(-1)^k x^{2k}}{(2k)!}∑k=0∞​(2k)!(−1)kx2k​ term by term?+

Where does the $(2n+1)!$ denominator come from after integrating $x^{2n}$ ?