Simply because the Maclaurin series is defined to be a Taylor series where a = 0.
Best example is that an "odd" (or "even") function's Maclaurin series only has terms with odd (or even) powers. cos(x) and sin(x) are examples of odd and even functions with easy to calculate Maclaurin series.
Yes. If the Maclaurin expansion of a function locally converges to the function, then you know the function is smooth. In addition, if the residual of the Maclaurin expansion converges to 0, the function is analytic.
A maclaurin series is an expansion of a function, into a summation of different powers of the variable, for example x is the variable in ex. The maclaurin series would give the exact answer to the function if the series was infinite but it is just an approximation. Examples can be found on the site linked below.
Normand MacLaurin was born in 1835.
Normand MacLaurin died in 1914.
Brian MacLaurin was born in 1949.
Neil MacLaurin was born in 1966.
The numerical value of pi is often found using a Taylor or Maclaurin series (Taylor series centered at 0).
John Maclaurin Dreghorn has written: 'The works of the late John Maclaurin, esq. of Dreghorn'
Colin Maclaurin died on 1746-06-14.
James Scott Maclaurin died in 1939.