Number series

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Problem: Examine the convergence of the series. 1) \( \sum_{n=1}^{\infty} \frac{1}{2^{n}+1} \), 2) \( \sum_{n=2}^{\infty} \frac{2 n}{\sqrt{n^{2}-1}} \).

2.10.21 Number series

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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

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

2.10.37 Number series

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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

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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

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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

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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

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