Number series

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Problem: Calculate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{3}{n(n-1)(n+3)} \]

2.10.31 Number series

1.25 $

Problem: Examine the series for convergence: \[ \sum_{n=1}^{\infty} \frac{1}{n \sqrt{(\ln n+2)^{5}}} . \]

2.10.32 Number series

1.25 $

Problem: 1. Write the first 5 terms of the series: \[ \sum_{n=0}^{\infty} \frac{n^{3}}{n+3} \] 2. Write it as a series with a common term: \[ \frac{0}{2}+\frac{1}{3}+\frac{4}{4}+\frac{9}{5}+\frac{16}{6}+\cdots \]

2.10.33 Number series

0 $

Problem: Examine the convergence of the series and, in the case of convergence, calculate the sum of the series with an accuracy of 0,01 : \[ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{\sin \frac{1}{n}}{\sqrt{n+2}} \]

2.10.34 Number series

0.75 $

Problem: Calculate the sum: \[ \sum_{k=0}^{n}\left(-4 k^{2}+5 k+2+\frac{4 k+7}{k^{2}+3 k+2}\right) C_{n}^{k} . \]

2.10.35 Number series

4.99 $

Problem: Calculate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{7}{49 n^{2}+35 n-6} . \]

2.10.36 Number series

0.75 $

Problem: Examine the convergence of the series: \[ \sum_{n=1}^{\infty} \frac{n !}{(3 n) !} \text {. } \]

2.10.37 Number series

0 $

Problem: Calculate the sum of the series with an accuracy of \( \alpha=0,1 \) : \[ \sum_{n=0}^{\infty}\left(-\frac{2}{3}\right)^{n} \text {. } \]

2.10.38 Number series

0.75 $

Problem: Examine the convergence of the series and, in the case of convergence, find the sum of the series with an accuracy of 0,01 : \[ \sum_{n=1}^{\infty}(-1)^{n+1} n^{3} \tan ^{5} \frac{\pi}{n} . \]

2.10.39 Number series

1 $

Problem: Examine the convergence of the series and, in the case of convergence, find the sum of the series with an accuracy of 0,01 : \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} \cdot \tan ^{-1} \frac{1}{5 n}}{n !} \]

2.10.40 Number series

1 $

Problem: Examine the convergence of the series: \[ \sum_{n=0}^{\infty} \frac{2 n+n^{5}}{3 n^{7}-7} . \]

2.10.41 Number series

0 $

Problem: Examine the convergence of the series: \[ \sum_{n=0}^{\infty} \frac{n !}{n^{2}+1} . \]

2.10.42 Number series

0.5 $

Problem: Examine the convergence of the series: \[ \sum_{n=0}^{\infty}\left(\frac{3}{4 n+7}\right)^{n} \text {. } \]

2.10.43 Number series

0 $

Problem: Determine the type of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(n+2)^{n}} \text {. } \]

2.10.44 Number series

0 $

Problem: Determine the type of convergence of the series: \[ \sum_{n=1}^{\infty} \frac{(-1)^{n} 6 n}{2 n} \text {. } \]

2.10.45 Number series

0 $

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