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Level 635

Power Series as Functions, Taylor Series


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∑(from j to ∞) c_j(x-a)^j
Define a power series centred at a:
What is the "interval of convergence" of a power series? What is the radius over convergence?
In the power series ∑(from j to ∞)c_j (x-a)^j there exists real number R≥0 such that the series converges absolutely on the interval (a-R,a+R) and diverges when |x-a|>R.
State the main derivative/integral rules for power-series
f(x) - f(a) = integral(from a to x) of d/dt f(t) dt
Define addition on power series
Consider ∑(from j to ∞) c_j (x-a)^j and ∑(from j to ∞) b_j (x-a)^j which are convergent to f(x) and g(x), on I₁ and I₂ Then: ∑(from j to ∞) (c_j ± b_j) (x-a)^j conve…
Define multiplication on power series
Consider ∑(from j to ∞) c_j (x-a)^j and ∑(from j to ∞) b_j (x-a)^j which have radii of convergence R₁ and R₂
∑(from j to ∞) f^(j) (a)(x-a)^j/j!
Let f(x) be an infinitely differentiable function on the open interval a. Define the taylor series
What is a maclaurin series?
A taylor series with a = 0
Define f^(0) (x)
f^(0) (x)= f(x) ... it is the function without taking any derivatives
that is: f(x) = ∑(from j to ∞)c_j (x-a)^j
Write this in mathematical notation and state what it implies.
Define T_n(x) in the context of a Taylor Series
T_n (x) is the partial sum whose degree is n
Are T_n(x) and S_n(x) equal in the context of Taylor series?
No. S_n(x) is the sum of the first n terms. T_n(x) is the partial sum whose degree is n
R_n (x) = ∑(for j>n) f^(j) (a) (x-a)^j /j!
How is the nth degree remainder of a Taylor series calculated?
For each x₀ in the interval of convergence, there exists a point c_n between x and a such that:
State Taylor's Remainder Theorem with respect to the a Taylor series: ∑(from j to ∞) f^(j) (a) (x-a)^j/j! with radius of convergence R
When does a Taylor Series converge?
If lim(n→∞) for all x∈ interval I, R_n(x)→0,
State Taylor's Theorem
If: ∑(from j to ∞)f^(j) (a)(x-a)^j/j! is a taylor series of f(x) with radius of convergence R such that for each x in (a-R,a+R), lim(n→∞) |R_n| = 0
What do we know about lim(n→∞) |xⁿ/n!|
For any x, lim(n→∞) |xⁿ/n!| = 0
State Taylor's Inequality
|R_n(x₀)| ≤ K|x₀-a|ⁿ⁺¹/(n+1)!
∑(from n to ∞) xⁿ/n!
State the Maclaurin series of: e^x
∑(from n to ∞) (-1)ⁿ(x²ⁿ⁺¹/(2n+1)!)
State the Maclaurin series of: sin(x)
State the Maclaurin series of: cos(x)
∑(from n to ∞) (-1)ⁿ x²ⁿ/(2n)!
∑(from n to ∞) xⁿ
State the Maclaurin series of: 1/(1-x)
State the binomial theorem:
(a+b)ⁿ = ∑(from k to n) (n choose k)
f(c)+f'(c)(x-c)/1!+[f''(c)(x-c)^2]/2!+[f'''(c)(x-c)^3]/3!+...+[f^(n)(c)(x-c)^n]/n!+[f^(n+1)(Zn)(x-c)^(n+1)]/(n+1)!
General Taylor Series
sin(x) at c=0
x-(x^3)/3!+(x^5)/5!-(x^7)/7!+(x^9)/9!+......+term in x^n+[+/-sin(Zn) or +/-cos(Zn)](x)^(n+1)/(n+1)!
cos(x) at c=0
1-(x^2)/(2!) +(x^4)/(4!)-(x^6)/(6!)+....+term in x^n+[+/-sin(Zn) or +/-cos(Zn)](x)^(n+1)/(n+1)!
1-x+(x^2)-(x^3)+(x^4)-(x^5)+.....
1/(1+x) at c=0
1+x+(x^2)+(x^3)+(x^4)+(x^5)+.....
1/(1-x) at c=0
x-(x^2)/(2)+(x^3)/(3)-(x^4)/(4)+(x^5)/(5)+.......
ln(1+x) at c=0
(x-1)-1/2(x-1)^(2)+1/3(x-1)^(3)-1/4(x-1)^(4)+......
ln(x) at c=1
1+x+(x^2)/2!+(x^3)/3!+(x^4)/4!+...+(x^n)/n!+(e^Zn)(x)^(n+1)/(n+1)!
e^x at c=0
1/(1-x)
x^n
1/(1+x)
(-1)^n*x^n
e^x
x^n/n!
sinx
((-1)^n*x^(2n+1))/(2n+1)!
cosx
((-1)^n*x^2n)/(2n)!
((-1)^(n-1)*x^n)/n
ln(1+x) from n = 1
ln((1+x)/(1-x))
2* sum of x^(2n+1)/(2n+1)
arctanx
((-1)^n*x^(2n+1))/(2n+1)
1/x
1 - (x-1) + (x-1)² - (x-1)³ ...
1/1+x
1 - x + x² - x³ ...
ln(x)
(x-1) - (x-1)²/2 + (x-1)³/3 - (x-1)⁴/4 ...
e^x
1 + x + x²/2! + x³/3! ...
sin(x)
x - x³/3! + x⁵/5! - x⁷/7! ...
cos(x)
1 - x²/2! + x⁴/4! - x⁶/6! ...
arctan(x)
x - x³/3 + x⁵/5 - x⁷/7 ...
sinh(x)
x + x³/3! + x⁵/5! + x⁷/7! ...
1+x+x²+x³+...xⁿ
1/(1-x)
1-x+x²-x³+...(-1)ⁿxⁿ
1/(1+x)
1-x²/2+x³/3+...(-1)ⁿxⁿ/n
ln(1+x)
tan⁻¹(x)
x-x³/3+x⁵/5+... (-1)ⁿx²ⁿ⁺¹/(2n+1)
sin(x)
x-x³/3!+x⁵/5!+... (-1)ⁿx²ⁿ⁺¹/(2n+1)!
1-x²/2!+x⁴/4!+...(-1)ⁿx²ⁿ/(2n)!
cos(x)
e^x
1+x+x²/2+x³/3!+... xⁿ/n!
cos(x)
1-(x^2/2!)+(x^4/4!)
sin(x)
1-(x^3/3!)+(x^5/5!)
ln(x)
(x-1)-((x-1)^2/2)+((x-1)^3/3)
ln(x+1)
x-(x^2/2)+(x^3/3)
1/(1-x)
1+x+x^2+x^3
1/(1+x)
1-x+x^2-x^3
e^x
1+(x/1!)+(x^2/2!)+(x^3/3!)
∑(x^n)/n!
e^x sigma
∑x^n
1/(1-x) sigma
ln(1+x) sigma
∑(-1)^(n-1) * x^n/n
sin(x) sigma
∑(-1)^n * x^(2n+1)/(2n+1)!
cos(x) sigma
∑(-1)^n * x^(2n)/(2n)!
arctan(x) sigma
∑(-1)^n * x^(2n+1)/(2n+1)
1/(1+x) sigma
∑(-1)^n * x^n
e^x
1 + x + x^2/2! + x^3/3!
sin(x)
x - x^3/3! + x^5/5! - x^7/7!
cos(x)
1 - x^2/2! + x^4/4! - x^6/6!
ln(1+x)
x - x^2/2 + x^3/3 - x^4/4
arctan(x)
x - x^3/3 + x^5/5 - x^7/7
1/(1-x)
1 + x + x^2 + x^3
1/(1+x)
1 - x + x^2 - x^3