Series

Question 1

What are the two summation formulas that must be memorised and the two that are given in the formula book?

Answer 1

Not Given:
Σⁿ_r=1(1) = n
Σⁿ_r=1(r) = 1/2 n(n+1)

Given:
Σⁿ_r=1(r²) = 1/6 n(n+1)(2n+1)

Σⁿ_r=1(r³) = 1/4 n²(n+1)²

Question 2

What are important summation results?

Answer 2

Σkƒ(x) = kΣƒ(x)
Σƒ(x)+g(x) = Σƒ(x) + Σg(x)

Question 3

How can you derive the formulas for other sums of series?

Answer 3

Using the given sums of series and using the results of series we can derive new sums of series

Question 4

How do you deal with using a given sum of series when the top of the sigma isn’t an n?

Answer 4

Swap all the n’s in the given sums of series for whatever is on top of the sigma

Question 5

How do you deal with a series where the bottom is not r = 1? e.g Σⁿ_r=k

Answer 5

You split the sum into two sums e.g
Σⁿ_r=k = Σⁿ_r=1 - Σ^k-1_r=1

Question 6

How do you solve a method of differences question?

Answer 6

• Make sure the term you are finding the sum of is in partial fraction form
• Write out the sum of the partial fractions, in a column where each row r=1,2,3…..n-2,n-1,n until you see a pattern
• Cross out all the terms that cancel (if for e.g the first three terms cancel then the last three should aswell)
• Once you have cancelled everything then write your short sum out and rearrange for the form they ask for

Note it may ask you to sub in a value of n or to split the sum into two sums as r≠1

Question 7

What is the formula for mclaurin series given in the formula book?

Answer 7

f(x) = f(0) + xf’(0) + (x²/2!)f ‘‘(0) + …

Question 8

What are the steps to finding a mclaurin series?

Answer 8

• State f(x) then find f(0)
• State f’(x) then find f’(0)
• Repeat this process until you have your first k non zero terms
• Sub into the mclaurin series formula and simplify

Note generally it will ask for the first k non zero terms

Question 9

How do you find the general sequence of a mclaurin expression in terms of r?

Answer 9

• Work out the nth term of the powers in the form ar+b
• If the first term is positive then the top of your expression must be (-1)^r+1, if your first terms negative the top of your expression is (-1)^r
• The bottom of your expression is generally the nth term of the factorials which is the same as the nth term of the powers (ar+b) to the factorial (ar+b)!

Your final expression will probably look like this:
(-1)^r+1/(cr+d)! * x^ar+b

Note it may help to leave some things unsimplified
Note the first term is always r=1

Question 10

What taylor series expansions are given in the formula book?

Answer 10

e^x
ln(1+x)
sin(x)
cos(x)
arctan(x)

Note the validity will change from expression to expression so you must work it out