Y2, C2 - Series

Question 1

What is the method of differences

Answer 1

If Un = f(n) - f(n + 1) then
The sum of Ur from r = 1 to n = f(1) - f(n + 1)

Question 2

How would you simplify (1 - 1/4)(1 - 1/9)(1 - 1/16)…(1 - 1/n^2)

Answer 2

Difference of two squares
= (1 - 1/2)(1 + 1/2)(1 + 1/3)(1 - 1/3) … (1 - 1/n)(1 + 1/n)
= 1/2 * (n+1)/n = (n+1)/2n

Question 3

How would you solve a generic method of differences question

Answer 3

1) Sub in values for r
2) Find a pattern and cancel out terms to simplify

Question 4

How would you solve a method of differences with impartial fractions

Answer 4

Solve the impartial fractions and then solve normally

Question 5

If Un = f(r) - f(r + 2), what is the sum of r equal to

Answer 5

Sum of Ur from r = 1 to n = f(1) + f(2) - f(n+1) - f(n+2)

Question 6

When subbing in values for a method of differences, how far should you go

Answer 6

To n

Question 7

How would you find the Maclaurin series for sinx

Answer 7

Write out f(x), f’(x), f’‘(x), etc…
From this write out f(0), f’(0), f’‘(0), etc…
Then sub into the Maclaurin expansion on formula sheet

Question 8

For what values of x is a Mclaurin expansion valid (for some functions on formula booklet (ln(1+x), arctanx))

Answer 8

-1 < x <= 1

Question 9

When finding a Mcalaurin expansion estimate for ln(1.05), what is our x value

Answer 9

0.05
ln(1 + x) = expansion
therefore ln(1 + 0.05) = expansion

Question 10

Rewrite lnl (root(1+2x)) / (1-3x) l so it can be expanded using the Mclaurin expansion

Answer 10

0.5ln(1+2x) - ln(1-3x)

Question 11

What is the nth derivative of 0 Mclaurin expansion

Answer 11

f^n(0) = n!an

Question 12

How to find Mclaurin expansion for cos(2x^2)

Answer 12

Plug 2x^2 in as x for the expansion of cos (x)

Question 13

For what range is ln( (root(1 + 2x)) / (1 - 3x) ) valid

Answer 13

1/2 * ln(1 + 2x) valid for -1/2 < x <= 1/2
ln(1 - 3x) valid for -1/3 <= x < 1/3
Therefore both valid for -1/3 <= x < 1/3

Question 14

How to find the Mclaurin expansion of ln(2 + 3x)

Answer 14

MUST be in form ln(1 + ax)
so ln(2 ( 1 + 3x/2)) = ln2 + ln(1 + 3x/2)
Find Mclaurin of ln(1 + 3x/2) and add ln2

Question 15

f’‘(x) = 4f’(x) - 5f(x). What is the third derivative of f(x)

Answer 15

f’’‘(x) = 4f’‘(x) - 5f’(x)