Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for and .
i) The Maclaurin series
ii) The radius of convergence of
i) For a power series , there is a positive number such that the series converges if and diverges if , this number is called the radius of convergence and the series is centred at .
ii) The Maclaurin series expansion for ,
iii) The ratio test:
If then the series is absolutely convergent, and
if then the series diverges.
The radius of convergence by ratio test is denoted as .
It is given that,
Replace by in the Maclaurin series expansion for .
This is the Maclaurin series expansion for
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