sequences and series

Question 1

how do you test if a sequence converges

Answer 1

• divide everything by the largest power of n in the denominator

Question 2

do constant sequences converge e.g. 1002

Answer 2

yes they converge at their constant in this case at 1002

Question 3

partial fraction method to test if a series converges

Answer 3

split the sequence into partial fractions

sub this back into the series

Evaluate the series with values and cancel terms

you are left with a fraction which is the sum of the series

determine the limit of the sum

Question 4

when does a geometric series converge / diverge

Answer 4

it converges when the ratio is less then 1

sum for convergence is a/1-r

it diverges is ratio is greater or equal to 1

Question 5

when does a harmonic series converge / diverge

Answer 5

when p is greater than 1 the series converges

when p is less than or equal to 1 the series diverges

Question 6

how to evaluate the nth term test

Answer 6

if the sequence diverges to a value that is not 0 then the series divergent

if the sequence converges to 0 the nth term test is inconclusive

Question 7

how to evaluate the alternating series test

Answer 7

• the sequence must be decreasing as the value of n increases
• if the sequence converges at 0 then the alternating series converges

Question 8

how to evaluate the comparison test

Answer 8

• both sequences must be positive
• manipulate the sequence to something you can evaluate
• evaluate the sequence if it converges the series converges

Question 9

how to evaluate absolute convergence

Answer 9

change the sequence to absolute value

evaluate the sequence

if it converges then the series has absolute convergence

Question 10

evaluate the ratio test

Answer 10

• take an+1/an
• simplify this and take the limit
• if L is less than 1 the series absolutely converges
• if L is more than 1 or doesn’t exist then the series diverges
• if L=1 the test is inconclusive

Question 11

how to evaluate a power series

Answer 11

• use the ratio test
• if limit is zero you can conclude the series converges for all x is R
• set L < 1
• solve the limit for x
• get two x values
• substitute these values into the original series and add the zero term onto the front of the series
• evaluate if they converge or not
• if the value converges then write it with a square bracket ] if it diverges leave it as a round bracket )

Question 12

how evaluate the Taylor series expansion

Answer 12

• find the derivatives of the function
• substitute the value given into the function
• with these values sub them into Taylor series