Kuis Deret Taylor dengan Jawaban (Polinom, ROC, Galat)

Question 1

Tentukan deret Maclaurin dari $\ln(1+x)$ (sebagai deret pangkat).

Accepted Answer

$\displaystyle \sum_{n=0}^{\infty} (-1)^n \frac{x^{n+1}}{n+1}$

Answer

$\displaystyle \sum_{n=0}^{\infty} (-1)^n \frac{x^n}{n+1}$

Answer

$\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{n}$

Answer

$\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{n-1}}{n}$

Question 2

Tentukan deret Maclaurin untuk $\arctan x$.

Accepted Answer

$\displaystyle \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1}$

Answer

$\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1}$

Answer

$\displaystyle \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{2n+1}$

Answer

$\displaystyle \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1}$

Question 3

Gunakan teorema Taylor untuk membatasi galat saat mendekati $\ln(1.2)$ dengan polinom Maclaurin derajat 3 dari $\ln(1+x)$ (yaitu pada $x=0.2$).

Accepted Answer

$\displaystyle |R_3(0.2)| \le \frac{6(0.2)^4}{4!}$

Answer

$\displaystyle |R_3(0.2)| \le \frac{(0.2)^4}{4}$

Answer

$\displaystyle |R_3(0.2)| \le \frac{(0.2)^3}{3!}$

Answer

$\displaystyle |R_3(0.2)| \le \frac{6(0.2)^4}{3!}$

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