Sequences/Series
Convergence Tests
Convergence Tests Pt. 2
Power Series
Taylor/Maclaurin Series
100
Determine whether the sequence is convergent or divergent. If convergent what does is converge to?
a_n=(2+n^3)/(1+2n^3)
Converges to
1/2
100
What are the requirements for integral comparison test
f(x) must be positive continuous and decreasing
100
Sigma_(n=1)^oo(3n^2+2)/(sqrt(4n^4+4))
Diverges by Divergence Test
100
Determine R and IOC
Sigma_(n=0)^oo(x^n)/(n!)
R=oo, I=(-oo,oo)
100
Use known Macluarin series to write as a Maclaurin Series
f(x)=sin(pix)
200
Determine convergence:
Sigma_(n=1)^oo 2^(2n+1)/5^n
converges to
8
200
Sigma_(n=1)^oo((n^2+1)/(2n^2+1))^n
Converges by Root Test
200
Sigma_(n=1)^oo((-2)^n)/(n^2)
Diverges by Ratio or Root Test
200
Determine ROC and IOC
Sigma_(n=1)^oox^n/(2n-1)
R=1, IOC=(-1,1]
200
Use known Maclaurin Series to obtain a maclaurin series and it ROC
f(x)=e^x+e^(2x)
300
Determine convergence
2+0.5+0.125+0.03125+...
Converges to
8/3
300
Sigma_(n=1)^oo (n^2+1)/(n^3+1)
Diverges by Limit Comparison Test
300
Sigma_(n=1)^oo(n!)/(100^n)
Diverges by ratio test
300
Find a power Series rep. and find IOC
f(x)=2/(3-x)
2/3 Sigma_(n=0)^oo(x/3)^n
R=3, I=(-3,3)
300
Find the Taylor Series and Radius
f(x)=e^(2x), a=3
R=oo
400
Determine convergence
Sigma_(n=1)^oo 1/(n(n+3))
Converges to
11/18
400
Sigma_(n=1)^oo(cos^2(n))/(n^2+1)
Converges by Comparison Test
400
Sigma_(n=1)^oo((-1)^(n-1))/(n^(1/3))
Converges Conditionally
400
Find power series representation and determine the IOC:
f(x)=x/(9+x^2)
I=(-3,3)
400
Find taylor series centered at a
f(x)=cosx, a=pi
R=oo
500
Determine Convergence
Sigma_(n=1)^oo (1+2^n)/3^n
Convergence to
5/2
500
Sigma_(n=1)^oo1/(nln(n))
Diverges by integral Test
500
Sigma_(n=2)^oo((-1)^nsqrtn)/(ln(n))
Fails AST b/c lim diverges to infinity. Diverges by divergence test.
500
Find power series representation and determine the IOC:
f(x)=x/(2x^2+1)
I=(-1/sqrt2,1/sqrt2)
500
f(x)=ln(x), a=2
R=2