Power series expansions can also be very useful for calculating indeterminate forms. Just for fun, try to calculate the indeterminate forms sin(x) lim et and lim x→0 - x 1 x2 using our power series representation of sin(x) and e". sin(x) = > (-1)"x2n+1 (2n + 1)! x' +. 7! 3! 5! n=0 x2 = 1+ x + 2! x4 + 4! ... n! 3! + + ||

Q: Q5: a\. Let f(x) = x³ - 3x² -1, x ≥ 2. State and prove the theorem of derivative of inverse…

A: Here , f(x) = x3-3x2-1 We find required value.

Q: Use the power series expansion of ƒ(x) = 1 x)²· 1- x for x < 1 to find a power series ex

A: Remember power series expansion for f(x)=11-x; x&lt;1 is  f(x)=11-x=∑k=0∞xk

Q: 4) (a) Find a degree 3 power series for the function sin(x) centered at I fla) = f (a) + f(a) (x-a)+…

A: By using degree-radian conversation

Q: <)=) k=0 x²k+1 (2k + 1)! For the x = value of the whose series expansion is given on the side, first…

A:

Q: If f(x) = 4,0 <x < 2n Find the Fourier sine series of f(x) A) Fes = En=1 [(-1)" – 1] sin () S %3D1 2…

A:

Q: f(x) = cos(x) and its MacLauren Series representation cos(x) (Α) /1 = 00 Σ ww. Use the Power Series…

A: f(x)=cos(x)      =∑n=1∞(-1)nx2n(2n)! We need to find the Maclauren series of  f(x)=x4cos(x)

Q: If f(x) = = ∞ n=0 n ∞ -2” and g(x) = Ỹ ( − 1)" mæª, find the power series of(f(x) – g(x)). n n=0 Σ…

A: The objective of the question is evaluate the sum of two series.

Q: +00 (-1)" The Maclaurin series expansion of tan-1I is given by tan¬ I = (2n + 1) 2n+1 for r €…

A:

Q: If f(x) = 3 n = 0 - and g(x) = (-1)^x, find the power series of(f(x) + g(x)) and of 1⁄(f(x) − g(x)).…

A: We are given the series       f(x) =  ∑n=0∞xnn!         g(x) =  ∑n=0∞(−1)nxnn!

Q: sinx x0 e²x-1 5. Use power series to evaluate lim

A: Power series expansion

Q: If f(x) = 2x" and g(x) Σ na", find the power series of f(r)g(x). %3D n=0 00

A:

Q: ο Σπo (-6)" x2n+3 Ο Σo 6" x2n+3 100 ο Σο (-6)" man Ο Σπο (-6)" an Ο Σπο 6 x 2n Find a power series…

A:

Q: Σ (-1)". It is known that for all x E R, COS x = (2n)! n=0 (a) Find a Power Series Representation…

A:

Q: Find Fourier series for f (x) = 2x + 1

A:

Q: Find the power series expansion of the function f(x) = = °Σ (−1)"ant1n ΟΣ (1)"2" min+1 - Σ*o(1)" r +…

A:

Q: (a) Write out the power series for 1 f(2) = zexp(iz?) + 1+ 2iz at zero.

A: The function is fz=z expiz2+11+2iz. As the power series is ex=1+x+x22!+x33!+…+xnn!+… at a = 0, then…

Q: (- 1)^2n- x2n-1 is 2 n = 0 (- 1)"x2n The derivative of the series 2 n= 0 2" 2" O True

A: We have to find the derivative of the given power series and need to verify whether the given…

Q: The formula provided in the question is a power series representation for g(x)= e-x^2

A: Step 1:​The power series of g(x)=e−x2:e−x2=n=0∑∞​n!(−x2)n​​Step 2:​e−x2=n=0∑∞​n!(−x2)n​Expanding the…

Q: 1. A power series is a series of the form and Cn's are where x is a

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: .Find a sequence that diverges, but doesn't diverge to infieity (a) an = n+1 (b) bn %3D Vn +1 (c) Cn…

A:

Q: Why will the alternating series test NOT apply to Σn=1 (-1) sin ((2n+1)) n 2 -?

A: Step 1:The alternating series test (also known as the Leibniz test) can be used to determine the…

Q: (-1)" Σ (2n)! R ALL XER, COS = x2 cos (2x) Find power series representation for Then by…

A:

Q: please answer 3 and 4

A: