Show:
Geometric and Nth Term
Integral and P-Series
Comparison Tests
Alternating Series
Ratio and Root
100
Determine whether the series converges or diverges Σ(1+1/n)^n
What is Diverge, Limit as n approaches infinity is e, e does not equal 0,
100
Determine whether the series converges or diverges 1/n^(1/3)
What is Diverge, 1/3 < 1
100
Determine whether the series converges or diverges Σ1/((n^3+1)^(1/2))
What is Diverge, Use LCT to find that bn = 1/n, limit of series as n approaches infinity x 1/bn is positive and finite, bn is a harmonic series
100
Determine whether the series converges or diverges. if converges, determine absolutely or conditionally Σ(-1)^n/(2n+1)
What is converges conditionally, lim as n approaches infinity = 0, use integral test on 1/(2n+1), u = 2n+1, du = 2, integral diveges
100
Determine whether the series converges or diverges Σn^7/7^n
What is converge, use ratio test, (n+1)^7/7^(n+1) x (7^n)/n^7 , lim (n+1)^7/7n^7 = 0, 0<1
200
Determine whether the following series converge or diverge Σ5^n/3^(n+2)
What is Diverge, 5^n/3^n+9, r=5/3, 5/3>1
200
Determine whether series diverges or converges 2/2(2^(1/2))+2/3(3^(1/2))+2/4(4^(1/2))
What is Converge, 2/n(n^(1/2)), n x n^(1/2) = n^(3/2), 3/2>1,
200
Determine whether the series converges or diverges. if converges, determine absolutely or conditionally Σ(-1)^n x ln(n)/n
What is converges conditionally, lim as n approaches infinity = 0, use integral test, u = ln(n), du = 1/n, series diverges
200
Determine whether the series converges or diverges Σ(3n/n+2)^n
What is diverges, use root test, lim as n approaches infinity of 3n/n+2 =3, 3>1
300
Determine whether the series converges or diverges Σn^(1/3)/(n+1)^(1/3)
What is diverge, Use nth term to get value of 1
300
Determine whether series diverges or converges Σ3n^2 e^-n^3
What is converge, u = -n^3, du = -3n^2, lim as n approaches infinity = 1/e^8, 1/e^8 < 1
300
Determine whether the series converges or diverges Σ(3n^2+5n^4)(n+1)^-5
What is Diverges, use LCT, bn = 1/n, lim is positive and finite , bn is harmonic series
300
Determine whether the series converges or diverges. if converges, determine absolutely or conditionally Σ(-1)^(n+1)(n!)2^3n/(2n)!
What is converges absolutely, lim as n approaches infinity = 0, use ratio test to get 8(n+1)/(2n+1)(2n+2), lim = 0, 0 < 1
300
Determine whether the series converges or diverges Σ-n^n/n^3n
What is converges, use root test, -n/n^3, -1/n^2, lim as n approaches infinity is 0, 0<1
400
Determine whether the series converges or diverges Σ8n^3-6n^5/12n^4+9n^5
What is diverges, use nth term, lim = -2/3, 2/3 < 1
400
Write out the first three terms of the series and use the integral test to determine whether the series converges or diverges. Σ1/(n(sqrt(ln(n)))
What is 1/(2(sqrt(ln(2))), 1/(3(sqrt(ln(3))), 1/(4(sqrt(ln(4))) + ...; diverges, u=ln(n), du=1/n, integral from 2 to infinity = infinity
400
Determine whether the series converges or diverges Σ1/n(n^2+1)^(1/2)
What is converge, bn = 1/ n^2, lim is positive and finite, use p-series for bn, 2>1,
400
Determine whether the series converges or diverges. if converges, determine absolutely or conditionally Σ(-1)^n e^-n^2
What is converges absolutely, lim as n approaches infinity = 0, use dct, 1/e^n^2<1/e^n, Bigger converges, smaller converges
400
Determine whether the series converges or diverges. Σ(-1)^(n+1) n^2 5^(n+1)/7^n
What is converges, use alternating series, lim as n approaches infinity is 0, use ratio test, lim as n approaches infinity equals 5/7, 5/7 < 1
500
Determine whether the series converges or diverges Σ3+4^(n-1)/3^(2n)
What is converges, 3+4^(n-1)/9^n, 3/9^n+4^(n-1)/9^n, 3(1/9)^n + 1/4(4/9)^n, both converge because both 1/9 and 4/9 is less than 1
500
find that value of p that makes the series converge 1/(x*(lnx)^p
What is x>1, use integral test to find that series coverges when x >1, u = ln x, du = 1/x
500
Determine whether the series converges or diverges Σ1+sin(n!)/n^2
What is converge, 1/n^2 + sin(n!)/n^2, 1/n^2 converges by p-series, sin(n!)/n^2 converges by DCT or ratio
500
let f be a positive, continuous, and decreasing function for x is greater than and equal to 1, such that an = f(n). it the series an coverges to S, then the remainder Rn = S - Sn is bounded by 0 < Rn < integral of f(x) from N to infinity. Find N such that Rn < 0.001 for the convergent series. Σ1/n^2+1
What is N greater than or equal to 1004, Rn is less than or equal to integral of 1/x^2+1 from N to infinity, integral is arctanx from N to infinity, artan(infinity) - (arctanN), pi/2 - artanN <0.001, arctan > 1.569
500
Determine whether the series converges or diverges. Σn^2/(2n-1)!
What is converges, lim (n+1)^2/(2(n+1)-1)! x (2n-1)!/n^2, lim (n+1)^2/(2n+1)(2n)(n^2) = 0, 0<1