Number series

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

2.10.16 Number series

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

2.10.17 Number series

0.75 $

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

2.10.18 Number series

0.75 $

Problem: Examine the series for convergence. \[ \sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n^{4}}\right) \text {. } \]

2.10.19 Number series

1 $

Problem: Examine the convergence of the series. \[ \sum_{n=1}^{\infty}(-1)^{n-1} \cdot \frac{(n-1)^{3}}{n^{4}+3 n^{2}+2} \]

2.10.20 Number series

1.25 $

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: Examine the convergence of the series. 1) \( \sum_{n=1}^{\infty}\left(1-\cos \frac{1}{\sqrt{n}}\right) \), 2) \( \sum_{n=1}^{\infty}(-1)^{n+1} \cdot\left(\frac{3 n}{3 n+1}\right)^{n^{2}} \).

2.10.22 Number series

1.25 $

Problem: Calculate the sum of the series. \[ \sum_{n=0}^{\infty}(-1)^{n} \cdot\left(\frac{3}{4}\right)^{n-1} . \]

2.10.23 Number series

0.5 $

Problem: Examine the series for convergence. 1) \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \cdot \sin ^{3} \frac{1}{n} \),2) \( \sum_{n=2}^{\infty}(-1)^{n} \frac{5 n}{7 n-4} \), 3) \( \sum_{n=1}^{\infty} \frac{n !}{4^{n} \cdot n^{n}} \),4) \( \sum_{n=1}^{\infty}(-1)^{n} \cdot \frac{e^{\frac{1}{n^{2}}}-1}{3 n^{2}} \), 5) \( \sum_{n=1}^{\infty} \cos ^{n} \frac{2 n}{2^{n} \sqrt{2 n+3}} \).

2.10.24 Number series

2.5 $

Problem: Examine the series for convergence: 1) \( \sum_{n=1}^{\infty}(-1)^{n} \cdot\left(\frac{3 n+5}{6 n+7}\right)^{n} \),2) \( \sum_{n=1}^{\infty}(-1)^{n} \frac{3^{-\sqrt{2 n+5}}}{\sqrt{2 n+5}} \).

2.10.25 Number series

1.25 $

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

2.10.26 Number series

0.75 $

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

2.10.27 Number series

1 $

Problem: Examine the series for convergence: 1) \( \sum_{n=1}^{\infty} \frac{2^{n+1}}{n^{2} \cdot 5^{n}} \), 2) \( \sum_{n=1}^{\infty}\left(\frac{n+1}{3 n+2}\right)^{n} \), 3) \( \sum_{n=1}^{\infty} \frac{1}{(n+2) \ln ^{2}(n+2)} \), 4) \( \sum_{n=1}^{\infty}(-1)^{n} \sin \frac{1}{n \sqrt{n}} \).

2.10.28 Number series

2.5 $

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

2.10.29 Number series

2 $

Problem: Calculate the sum of the series: \[ \sum_{n=1}^{\infty} \frac{12}{36 n^{2}-12 n-35} . \]

2.10.30 Number series

1 $

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