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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