Power Series (Maclaurin and Taylor) Flashcards
Taylor series formula:
Sum from n=0 to infinity of:
f^n(x_o)/n! * (x-x_o)^n
Where initial is value of derivative at x_o, a given value centred around.
Maclurin series:
Taylor series centred around 0 so x_o = 0
We may have to give the general form for the series how do we do this:
Have sum, then largely logic, value compared to n, alternating? Does it skip etc.
How do we work with composite taylor series
We write our main one then substitute in values and expand when necessary.
Maclaurin series for: e^x
1+ x + x^2/2! +x^3/3! +x^4/4! ….
Maclaurin series for: sin(x)
x-x^3/3!+x^5/5!-x^7/7!….
Maclaurin series for: cos(x)
1-x^2/2! +x^4/4! - x^6/6!….
Maclaurin series for: tan^-1(x)
x-x^3/3+x^5/5-x^7/7….
Maclaurin series for: 1/1-x
1+x+x^2+x^3+x^4…
Maclaurin series for: ln(1+x)
x-x^2/2+x^3/3-x^4/4….
Convergence/ divergence tests in order of use (ish)
initial limit test
integral test
ratio test
if obviously alternating use alternating series test
initial limit test
evaluate limit of the function as n approaches infinity, if it doesn’t equal 0 then the series diverges
integral test
For a decreasing, and continuous over the range,
we integrate the function between 0 and infinity, if gives a real number then it converges.
ratio test
limit of n approaching infinity of |an+1/an| = L
If L>1 diverges, L<1 converges and if L=1 inconclusive
alternating series test
If we can split the series into (-1)^n or n+1 * bn then we can evaluate the bn as long as it is greater than or equal to 0 an bn=>bn+1 then if we evaluate bn at infinty and it = 0 then an converges
