Functional sequences and series

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Problem: Find the sums of the following series: \[ \sum_{n=1}^{\infty} \frac{x^{n+1}}{\left(1-x^{n}\right)\left(1-x^{n+1}\right)} \] if a) \( |x|<1 \); b) \( |x|>1 \).

2.9.1 Functional sequences and series

2.03 $

Problem: Find the domain of convergence of the functional series: \[ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{4}} \sin ^{n}(3 x) \text {. } \]

2.9.2 Functional sequences and series

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Problem: Find the domain of convergence of the functional series: \[ \sum_{n=1}^{\infty}\left(1-4 x^{2}\right)^{n} \text {. } \]

2.9.3 Functional sequences and series

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Problem: Does function \( f_{n}(x)=\cos \frac{x}{n} \cdot \cos x \) uniformly converge in a) \( [0, \pi] \), b) \( \mathbb{R} \).

2.9.4 Functional sequences and series

2.03 $