Finding Limit of Sequences
a_n = (nsin(n))/(n^2+1)
sequence converges to 0 due to the squeeze theorem
sum_(k=1)^infty1/(k^2+1)
series converges due to the LCT
Find the Infinite Sum:
sum_(k=2)^infty(-0.85)^k
s_infty=0.7225/(1+0.85)
find the radius of the series:
sum_(k=0)^infty((-3)^k(x)^k)/(sqrt(k+1))
radius = 1/3
Find the linear approximating polynomial for the following function centered at the given point a.
f(x)=3x^2
, a = 1
P_1(x)=6x-3
b_n=(n^2)/(n^3+1)
sequence converges to 0
sum_(k=1)^infty((-1)^k)/(sqrt(n+1))
series conditionally converges because it failed absolute value test but passed Alternating test
Find the Partial Sum:
sum_(k=1)^infty1/((k+6)(k+7))
S_k=k/(7k+49)
Given the radius = 1/20, find the interval of convergence:
sum_(k=0)^infty(20x)^k
(-1/20,1/20)
Using the function f(x) = e^x centered at a=0, use the linear polynomial to approximate -1/2
Linear approx: 1/2
sum_(k=4)^infty(e^(2n))/((n-2)!)
series converges absolutely due to the ratio test
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
Find the Taylor Polynomial of order n=2 for the following function centered at the given point a.
f(x) = -1/x, a = 1
p_2(x)=-x^2+3x-3