Hello tutors , I need help with this series word problem. I apologize it's long, I would break it up into parts but unfortunately, this uses each previous part as a reference. Please let me know how you can come across this, thank you!

The function 3 2 1 1 2 3 plays an extremely important role in probability and statistics, as the probability density function of the standard normal (Gaussian) distribution. This means that if a random variable X is normally distributed, then, for example, the probability that X is Within 1 standard deviation above its mean is 3 2 1 1 2 3 Unfortunately, and rather famously, the function p(.'E) is an example of a function that cannot be integrated "symbolically"; that is, its antiderivative is not an elementary function. Traditionally, before computers gave us other ways to do this, generations of students have had to approximate integrals like (1) by looking them up in huge tables, and these tables still ll up pages upon pages in statistics textbooks. In this problem, you'll explore one way to approximate an integral like (1) without needing more than a simple calculator (and which, in theory, you could compute with out a calculator). Indeed, the method given here is how many of the aforementioned tables were computed back in the days before computers. (a) 00) Write down the Taylor series for em, centered at 0. You should be able to write the entire thing using 2 notation. Then, in the expression you wrote down, substitute 1 2 $372 a: in place of m to get the Taylor series for 62 2 , centered at m = 0. Integrate the Taylor series you got in part (a), to get a Taylor series for _l2 /2wd$ Use the rst 5 or more terms of your result from part (b) to compute the de nite integral (1). (Don't forget to divide by m at the end!) Congratulations! You've now computed the probability that a Gaussian random variable is within 1 standard deviation above its mean! Try comparing your answer to a value from a table or a Google search to see how accurate your answer is. (If you did everything correctly, 5 or 6 terms of the Taylor series from part (b) should give you around 4 or 5 digits of accuracy.)