1429. Check that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$ converges. How many terms of this series need to be taken to compute its sum with an accuracy of 0.01?
Solution. For this series, all conditions of the Leibniz criterion are satisfied, so the series converges. We can write the sum of the series in the form: $S=S_{n}+r_{n},\left|r_{n}\right|<\left|a_{n+1}\right|$. We need to find such an $n$ that $\left|a_{n+1}\right|=\frac{1}{\sqrt{n+1}} \leqslant 0.01$. Solving the equation $\frac{1}{\sqrt{n+1}}=0.01$, we get $n=9999$, hence $\left|r_{9999}\right|<0.01$. Considering the conditions of the problem and this result, we find $0<r_{9999}<0.01$. Therefore, $S=S_{9999}+r_{9999}$, from which $S \approx S_{9999}$ with an accuracy of 0.01 (with a deficit). Note that this series converges very slowly, as do all other conditionally convergent series.