Series Tests

Question 1

nth-term

Answer 1

1. an
2. null
3. lim n->inf an != 0
4. Cannot be used to show convergence

Question 2

Geometric Series

Answer 2

1. ar^n
2. abs(r)<1
3. abs(r)>=1
4. Sum: S = a/(1-r)

Question 3

Telescoping Series

Answer 3

1. (bn-bn+1)
2. lim n->inf bn = L
3. null
4. Sum: S = b1 - L

Question 4

p-Series

Answer 4

1. 1/n^p
2. p > 1
3. 0 < p <= 1
4. null

Question 5

Alternating Series

Answer 5

1. (-1)^n-1 * an
2. 0<an+1<=an and lim n->inf an = 0
3. null
4. Remainder: abs(RN)<=aN+1

Question 6

Integral (f is continuous, positive, and decreasing)

Answer 6

1. an, an=f(n) >= 0
2. int(1, inf, f(x)) converges
3. int(1, inf, f(x)) diverges
4. Remainder: 0 < RN < int(N, inf, f(x))

Question 7

Root

Answer 7

1. an
2. lim n->inf nth-root(abs(an)) < 1
3. lim n->inf nth-root(abs(an)) > 1 or =inf
4. Test inconclusive if lim n->inf nth-root(abs(an)) = 1

Question 8

Direct Comparison (an, bn > 0)

Answer 8

1. an
2. 0<an<=bn and bn converges
3. 0<bn<=an and bn diverges
4. null

Question 9

Ratio

Answer 9

1. an
2. lim n->inf abs(an+1/an) < 1
3. lim n->inf abs(an+1/an) > 1 or =inf
4. Test inconclusive if lim n->inf abs(an+1/an) = 1

Question 10

Limit Comparison (an, bn > 0)

Answer 10

1. an
2. lim n->inf an/bn = L > 0 and bn converges
3. lim n-> inf an/bn = L > 0 and bn diverges
4. null