1019. Using appropriate series, calculate with an accuracy of $0.0001: 1) \ln 1.1 ; 2) \sqrt[4]{17}$.
Solution. For calculating approximate values of a function with a given accuracy, it is convenient to use series when the corresponding series is alternating; for an alternating convergent series, it is easy to estimate the error of the approximate value of the sum - it is less than the absolute value of the first 13 discarded terms (§ 2).

In other cases, approximate values of the function with a given accuracy are calculated using the Taylor (Maclaurin) formula, as shown in the solution of problem 299 (Chapter III, § 1).

1) Let's take the series for the function $\ln (1+x)$, obtained in the solution of problem 1012:

$$
\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\ldots+(-1)^{n-1} \frac{x^{n}}{n}+\ldots
$$

which converges to $\ln (1+x)$ in the interval $(-1 ; 1]$, and, setting $x=0.1$, we get a series for calculating $\ln 1.1$ with any accuracy:

$$
\ln 1.1=0.1-\frac{0.1^{2}}{2}+\frac{0.1^{3}}{3}-\frac{0.1^{4}}{4}+\ldots
$$

The absolute value of the fourth term of this series is less than 0.0001. Therefore, according to the property of an alternating convergent series (§ 2), to calculate the approximate value of $\ln 1.1$ with an accuracy of 0.0001, it is sufficient to take the sum of the first three terms of the series

$$
\begin{aligned}
& \ln 1.1 \approx 0.1-\frac{0.01}{2}+\frac{0.001}{3} \approx 0.0953 \\
& -360-
\end{aligned}
$$

2) Transform the given root $\sqrt[4]{17}=\sqrt[4]{16+1}=2\left(1+\frac{1}{16}\right)^{\frac{1}{4}}$ and apply the binomial series (B), obtained in the solution of problem 1012, setting $x=\frac{1}{16}, m=\frac{1}{4}$:

$$
2\left(1+\frac{1}{16}\right)^{\frac{1}{4}}=2\left[1+\frac{1}{4 \cdot 16}-\frac{1 \cdot 3}{4 \cdot 8 \cdot 16^{2}}+\frac{1 \cdot 3 \cdot 7}{4 \cdot 8 \cdot 12 \cdot 16^{3}}-\ldots\right]
$$

To determine how many of the first terms of this alternating convergent series to take for calculating $\sqrt[4]{17}$ with an accuracy of 0.0001, we calculate several consecutive first terms of the series: $a_{1}=1 ; a_{2} \approx 0.01562 ; a_{3} \approx-0.00037 ; a_{4} \approx 0.00001$.

According to the property of an alternating convergent series, if we limit ourselves to the sum of the first three terms of the series, then the error of the desired approximate value of the root will be less than $2 a_{4} \approx 2 \cdot 0.00001<0.0001$. Therefore,

$$
\sqrt[4]{17} \approx 2(1+0.01562-0.00037) \approx 2.0305
$$