Series&Sequences
Perform an index shift so that the following series starts at n=3.
∞∑n=7(4−n)/(n^2+1)
∞∑n=3(−n)/((n+4)^2+1)
Determine if the following series converges or diverges.
∞∑n=2(1)/(2n+7)^3
Integral test = 1/484, so converges
Determine if the following series converges or diverges. Give your reasons related to the method.
∞∑n=1(−1)^(n−1)/(7+2n)
Converges by A.S.T.
For the following power series determine the interval and radius of convergence.
∞∑n=0(n+1)*(x−2)^n/(2n+1)!
Interval of convergence is −∞<x<∞ and the radius of convergence is R=∞ .
Given that ∞∑n=0(1)/(n^3+1)=1.6865, determine the value of ∞∑n=2(1)/(n^3+1).
0.1865
Determine if the following series converges or diverges. Give your reasons related to the method.
∞∑n=2(n−1)/(sqrt(n^6+1))
b(n) = 1/n^2, converges by p-series.
Determine if the following series converges or diverges. Give your reasons related to the method.
∞∑n=0(n^(1−3n))/(4^(2n))
L=0<1 and so by the Root Test the series converges.
For the following power series determine the interval and radius of convergence.
∞∑n=1(6^n*(4x−1)^(n−1))/n
Interval:5/24≤x<7/24, R=1/24
Determine if the series converges or diverges. If the series converges give its value.
∞∑n=1(3)/(n^2+7n+12)
converges to 3/4
Determine if the series converges or diverges. If the series converges give its value.
∞∑n=3(3)/(n^2−3n+2)
3*ln(2)
Determine if the following series converges or diverges. Give your reasons related to the method.
∞∑n=3(e^(4n))/(n−2)!
L = 0<1 and so by the Ratio Test the series converges.
For the following power series determine the interval and radius of convergence.
∞∑n=0(4^(1+2n)*(x+3)^n/(5^(n+1))
Interval:−53/16<x<−43/16, R=5/16
Find the sum.
∞∑n=1(9^(−n+2)*4^(n+1)
1296/5
Determine if the following series converges or diverges. Give your reasons related to the method.
∞∑n=1(2n^3+7)/(n^4*sin^2(n))
b(n) = 1/n, diverges by p-series.
Determine if the following series converges or diverges. Give your reasons related to the method.
∞∑n=4(−5)^(1+2n)/(2^(5n−3))
L=25/32<1 and so by the Root Test the series converges.
Give a power series representation for the following function.
h(x)=x^4/(9+x^2)
∞∑n=0(−1)^n*(1/9)^(n+1)*x^(2n+4)