Calculus II   
Mathematica Project 3
Due Date: December  13, 2012

1.	Detrmine the convergence or divergence of the given series. 

	a)	(∑_(n=1)^∞)(1/( 3n+1))		b)	(∑_(n=1)^∞)((((-1))^n)/( 3n+1))

2.	Suppose that f is positive, conitnuous and decreasing function for x≥1. Let (Subscript[a, n])=f(n). It can be shown that if the infinite series (∑_(n=1)^∞)(Subscript[a, n]) converges to S, then the remainder (Subscript[R, M])=S-(Subscript[S, M]) is bounded  by 
				
				0≤(Subscript[R, M])≤(∫_M^∞)f(x) d x. 

Use this fact to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. (Hint: The maximum error is the improper integral in the above inequality. You need to define (Subscript[S, M ])as S[M_]:=(∑_(n=1)^M)(Subscript[a, n]). Use N[S[M]] to get a numerical value of (Subscript[S, M]) and NIntegrate for the approximation of the error.)

a)	(∑_(n=1)^∞)  (1/(n^4))	100 terms.

b) 	(∑_(n=1)^∞)(1/(((n+1) [ln(n+1)])^3))	 1000 terms.

3.	Let  f(x)  =  e^sin(x)/(200  + 49 x^2). 

		a)	Find  the Taylor polynomials (Subscript[P, n])(x) of f at x =0 for n = 1, 2, ... 6.
		b) 	Draw the graphs of the function f and its Taylors Polynomials found in (a)
		c)	Use the graph in (b) to estimate the radius of convergence of the Taylor series of f at x =0