Paper 1, Section II, F

Analysis I | Part IA, 2009

For each of the following series, determine for which real numbers xx it diverges, for which it converges, and for which it converges absolutely. Justify your answers briefly.

(a) n13+(sinx)nn(sinx)n\sum_{n \geqslant 1} \frac{3+(\sin x)^{n}}{n}(\sin x)^{n},

(b) n1sinxn(1)nn\quad \sum_{n \geqslant 1}|\sin x|^{n} \frac{(-1)^{n}}{\sqrt{n}},


n1sin(0.99sin(0.99sin(0.99x)))n times .\sum_{n \geqslant 1} \underbrace{\sin (0.99 \sin (0.99 \ldots \sin (0.99 x) \ldots))}_{n \text { times }} .

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