Нахождение рядов Тейлора или Маклорена для функции

The polynomial that never stops

• A Taylor polynomial stops at degree $n$. Let it run forever and you get a Taylor series 泰勒级数.
• $\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}$ — the infinite version, exactly representing $f$ where it converges.
• Centered at $0$, it's a Maclaurin series 麦克劳林级数.
• A handful of these are worth knowing by heart.

The must-know Maclaurin series

• $\displaystyle e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$
• $\displaystyle \sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$
• $\displaystyle \cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$
• $\displaystyle \frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}=1+x+x^2+\cdots$ (the geometric series, $|x|<1$).

Each extra Maclaurin term hugs the function over a wider range

Building one from scratch

• Compute $f(a),f'(a),f''(a),\dots$ and plug into $\frac{f^{(n)}(a)}{n!}(x-a)^n$, term by term.
• Look for the pattern in the derivatives to write the general $n$th term.
• The known series above are usually derived once then reused — memorize them.
• The general term is what makes it a series, not just a few terms.

$\sin$ and $\cos$ share a family

• Notice $\sin x$ uses odd powers and $\cos x$ uses even powers — and both alternate sign.
• $e^x$ has every power with all-positive coefficients $\tfrac1{n!}$.
• These patterns let you write out terms quickly and spot which series a problem wants.
• Recognizing the family is half the battle on the exam.