Unit 3: Convergence of Series and Taylor Polynomials

Finding Limit of Sequences

100

a_n = (nsin(n))/(n^2+1)

sequence converges to 0 due to the squeeze theorem

200

b_n=(n^2)/(n^3+1)

sequence converges to 0

Determining Divergence, Absolute, or Conditional

100

sum_(k=4)^infty(e^(2n))/((n-2)!)

series converges absolutely due to the ratio test

200

sum_(k=1)^infty1/(k^2+1)

series converges due to the LCT

300

sum_(k=1)^infty((-1)^k)/(sqrt(n+1))

series conditionally converges

Finding The Sum of a Series

100

sum_(k=1)^infty1/((k+6)(k+7))

s_n=n/(7n+49)

200

Suppose that you take 240 mg of an antibiotic every 4 hr. The half-life of the drug is 4 hr (the time it takes for half of the drug to be eliminated from your blood). Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood exactly. 

480 mg is the long-term amount of antibiotic in your blood

Finding The Radius/Interval of Convergence

100

sum_(k=0)^infty((-3)^k(x)^k)/(sqrt(k+1))

radius = 1/3

Interval = (-1/3,1/3]

Taylor Polynomial (Very Scary)

100

Find the linear approximating polynomial for the following function centered at the given point a. Find the quadratic approximating polynomial for the following function centered at the given point a. Use the polynomials to approximate the given quantity. 

f(x) = -1/x, a = 1; approx: -1/0.92

p_1(x)=x-2

p_2(x)=-x^2+3x-3

approx-1.08

approx-1.0864