Sequence & Series Test

Question 1

Con/div of a sequence {}

Answer 1

Take limit of a_n
* if limit is a finite number, sequence converges to that number
* if limit is infinite, series diverges

Question 2

Divergence Test

Answer 2

Take the limit of a_n if it DNE or is equal to zero then the series diverges

Question 3

Geometric Series Test

Answer 3

Geometric series of form arⁿ:
* if |r|>=1 the series diverges
* if |r|<1 the series converges and the sum of the series is ar^l/1-r

Question 4

Integral Test

Answer 4

Conditions of f(x):
1. continuous
2. positive
3. decreasing
If the limit of the integral of f(x):
* exists, the series converges
* does not exist, the series diverges

Question 5

P-series Test

Answer 5

P-series of form 1/n^p:
* converges if p>1
* diverges if p<=1

when p=1 is called harmonic series

Question 6

Limit Comparison Test

Answer 6

b_n is a similar function (simple version)
Assume: a_n, b_n
Let L = limit of a_n/b_n
1. if 0 < L < infinity AND the series b_n converges -> series a_n converges
2. if 0 < L < infinity AND the series b_n diverges -> series a_n diverges
3. if L = 0 AND series b_n converges -> series a_n converges
4. if L = infinity and series b_n diverges -> series a_n diverges

Any other case is inconclusive and you must use another test

Question 7

Alternating Series Test

Answer 7

Alternating series of form (-1)ⁿb_n where b_n > 0:
if the limit of b_n is 0 AND {b_n} is decresing -> the alternating series converges

Question 8

Ratio Test

Answer 8

For a series a_n:
* if the limit of a_{n + 1}/a_n < 1 -> series a_n converges
* if the limit of a_{n + 1}/a_n > 1 or infinity -> series a_n diverges

If equal to 1 the test is inconclusive