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Students of a class were asked to find examples of Taylor
series from books and internet and submit interesting examples.\\
Following responses were received.
- $\ln x = \left({x-1\over x}\right)+{1\over2}\left({x-1\over x}\right)^2+{1\over3}\left({x-1\over x}\right)^3+\cdots$
- $\ln \left({1+x\over1-x}\right) = 2\left(x+{x^3\over3}+{x^5\over5}+{x^7\over7}+\cdots\right)$
- ${\rm cosec} x= {1\over x}+{x\over6}+{7x^3\over360}+{31x^5\over15210}+\cdots$
- ${\rm cosec}^{-1}x={1\over x}+{1\over2.3x^3} + {1.3\over2.4.5x^5}+{1.3.5\over2.4.6.7x^7}$
- $\ln |\sin x|=\ln |x|-{x^2\over6}-{x^4\over180}-{x^6\over2835} - \cdots$
Without trying to verify the correctness of expansion, give your
comment for each of the above responses paying attention to the
following. ``Is the example a correct response to the question
posed? If not why not?''
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