Number series

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Problem: Examine the indicated series with positive terms for convergence. 1. \( \sum_{n=1}^{\infty} \frac{n^{n}}{(n+1) !} \), 2. \( \sum_{n=1}^{\infty}\left(\frac{1}{\ln (n+1)}\right)^{2 n} \).

2.10.1 Number series

1.27 $

Problem: Examine the alternating series for convergence and absolute convergence: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(2 n+1)}{n} \text {. } \]

2.10.2 Number series

0.51 $

Problem: With an accuracy of \( \varepsilon=0.001 \) find the partial sum \( S \) of the series \[ S=\sum_{n=1}^{\infty}\left(\frac{4 n+7}{6 n+5}\right)^{n^{2}} \cdot / \]

2.10.3 Number series

3.81 $

Problem: With an accuracy of \( \varepsilon=0,001 \) find the partial sum \( S \) of the series \[ S=\sum_{n=1}^{\infty} \frac{(3 n+2)^{2}}{7^{n}} . \]

2.10.4 Number series

3.81 $

Problem: With an accuracy of \( \varepsilon=0,001 \) find the partial sum \( S \) of the series \[ S=\sum_{n=1}^{\infty} \frac{n}{(3 n+2)^{2}} \text {. } \]

2.10.5 Number series

3.81 $

Problem: Examine the series for the absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n+1} \cdot \frac{1}{3 n-1} \text {. } \]

2.10.6 Number series

0.51 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n+1} \cdot a_{n}, \quad \text { where } a_{n}=\frac{1 \cdot 4 \cdot \ldots \cdot(3 n-2)}{3 \cdot 5 \cdot \ldots \cdot(2 n+1)} \]

2.10.7 Number series

0.76 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n} \cdot \frac{\ln n}{n} . \]

2.10.8 Number series

0.76 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=3}^{\infty}(-1)^{n} \cdot \frac{1}{n \ln n(\ln \ln n)^{3}} . \]

2.10.9 Number series

1.27 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=1}^{\infty}(-1)^{n-1} \cdot \frac{n^{2}}{2^{n}} . \]

2.10.10 Number series

0.76 $

Problem: Examine the series for absolute and conditional convergence: \[ \sum_{n=3}^{\infty}(-1)^{n} \cdot \frac{1}{n \ln n \sqrt{\ln \ln n}} \]

2.10.11 Number series

1.27 $

Problem: Make sure that Leibniz test can't be applied to the series \( \sum_{n=1}^{\infty} N_{n} \) with the terms given below \( (k \in N) \). Examine this series for convergence in other ways. \[ a_{n}=\left\{\begin{array}{lc} \frac{1}{\sqrt{k+1}+1}, \quad n=2 k-1 \\ -\frac{1}{\sqrt{k+1}-1}, \quad n=2 k \end{array}\right. \]

2.10.12 Number series

1.27 $

Problem: Make sure that the Leibniz test cannot be applied to the series \( \sum_{n=1}^{\infty} N_{n} \) with the terms \( (k \in N) \) indicated below. Examine this series for convergence in other ways. \[ a_{n}=\left\{\begin{array}{cc} \frac{1}{3^{k}}, & n=2 k-1 \\ -\frac{1}{2^{k}}, \quad n=2 k \end{array}\right. \]

2.10.13 Number series

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Problem: Examine the series for convergence: 1. \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !} \), 2. \( \sum_{n=1}^{\infty} \frac{3 n+3}{4^{n}} \).

2.10.14 Number series

1.02 $

Problem: Examine the series for convergence. \[ \sum_{n=1}^{\infty}(-1)^{n} \cdot\left(\frac{3 n-2}{3 n+2}\right)^{n} \]

2.10.15 Number series

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