Infinite Series Tests

nth term test

1. take the limit of the series

2. if the limit approaches 0, the series test is inconclusive

3. if the limit approaches anything else, the series diverges

geometric series test

1. find r value

2. if r is less than one, the series converges

3. if r is greater than one, the series diverges

4. sum = a1 / (1-r)

p series test

1. 1 / (pⁿ)

2. if n is greater than 1, the series converges

3. if n is less than or equal to one, the series diverges

telescoping series

1. use if denominator is factorable, make partial fractions with cover up test

2. expand until the pattern is clear and most terms cancel out

3. add up remaining terms to find the sum

alternating series test

alternators: (-1)ⁿ, (-1)^n-1, cos(πn),

conditions: decreasing, an+1 < an

1. remove the alternator and take the limit of the leftover series

2. if the limit approaches 0, the series converges conditionally

3. if the limit does not approach 0, the series diverges by nth term test

direct comparison test

limit comparison test

conditions: finite, positive

1. find the limit of the ratio of the terms (multiply series by reciprocal of the comparison factor)

2. find a comparison factor by stripping down the original series

3. calculate the limit

4. if the limit is greater than 0, both series either converge or diverge together

5. if the limit is 0 and the comparison factor converges, the series will converge

6. if the limit is infinite and the comparison factor diverges, the series will diverge

root test

1. take the nth root of a series that’s raised to the power of n

2. take the limit of the root

3. if the limit is less than 1, the series converges absolutely

4. if the limit is greater than 1, the series diverges

5. if the limit is equal to 1, the test is inconclusive

integral test

conditions: positive, continuous, decreasing

1. if the series looks like u sub, take the improper integral of the series

2. if the integral is finite and approaches a real number, the series converges

3. if the integral is infinite and does not approach a real number, the series diverges

ratio test

1. take the limit of consecutive terms (aⁿ⁺¹) / (aⁿ)

2. (same as root test)

3. if the limit is less than 1, the series converges absolutely

4. if the limit is greater than 1, the series diverges

5. if the limit is equal to 1, the test is inconclusive