(a) Let P(n) be the statement that: e-/ lim =0 for any positive integern • ()Use L'Hopital's rule to proe that P(1) is true. • (ii) Use mathematical induction method to verify that P(n) is true.

Q: 3) Find Fourier series expansion of he funchion f(x) - T/2 Śn 2o %3D cos n

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Q: -2z + 3 Find all Taylor and Laurent series of f(z) : with center 0. 2 - 3z + 2

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Q: 6) Evaluate the limit using Taylor series. lim 1 Inx

A: Taylor Expansion of ln x about x=1 is as follow  :                 ln…

Q: Q5/ Find Fourier series of the Function: f(x) = x if (-n <x< n)

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Q: 1. Find the Fourier Series coefficients representation of X if -T <x < 0 f(x) TT – X if 0<x< T

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Q: 29-30 The limit of an indeterminate form as x x0 can some- times be found by expanding the functions…

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Q: a) Find the Fourier series of the function f defined by f(x) = 1 if −π < x < 0, f(x) = 0 if 0 < x <…

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

Q: =1 using Cauchy's Residue theorem.

A: Cauchy's Residue theorem: Let fz be analytic within and on a positively oriented simple closed…

Q: Example 2: Find the Fourier series -1, f (x) = 1, f«) = - 1< x < 0 0 <x < 1

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Q: 1 Find the Fourier series for 1, if -п <х <0 { { -1, f (x) = -1, if 0 < x< n

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Q: (3k + 8)! Consider the series> ar k3k k-1 k-1 a) What is ak+1? ak+1 = aktl ? b) What is ak ak-1/ak =…

A: The given series,     ∑k=1∞ak=∑k=1∞3k+8!k3k

Q: Determine the Fourier series of the function f (x) defined on the interval −1 ≤ x ≤ 1 using the…

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Q: (a). Determine the Fourier series for the 2p-periodic function given on the interval [0,2p) by f(x)=…

A: a. Hint: The Fourier transform of a function f(x) is…

Q: 6 + 7x (1 point) Let f(x) = Compute f(x) (6+7x)/x f(1) 13 f' (x) -6/(x^2) f'(1) -6 f" (x) 12/(x^3)…

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Q: Determine whether the Fourier series of the following functions converge uniformly or not. A) f(x)…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 3 1. Find the Fourier series on -≤x≤ for f(x) = sin ³ x, which is an odd function.

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Q: Find Fourier series of the Function: f(x) = x if (–n <x< n)

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Q: 2. Find the Taylor series of g(x) = at x = -1.

A: 2. g(x)=1x5 at x=-1

Q: 2n²+3n 2 3) Prove (with an e - N proof): lim n-0o 3n2+8n+2 3

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Q: 1. Find the Fourier series of the function -x if -4 <x<0 f(x) = S(x + 8) = f(x) if 0<x<4

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Q: 1. This exercise will guide you through the computation of the following limit et – 1- x lim a→0 3x…

A: Using Taylor expansion

Q: Find Fourier Series of the Function: -K f(x) = {-k -π<x<0 0 < x < π K

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Q: 1)Determine whether the Fourier series of the following functions converge uniformly or not.  f(x)…

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Q: a. Find the Taylor series generated by f if f(x) = x* + x² + 1 %3D ,a = -2

A: Taylor series of function is given by f(x)=f(a)+(x-a)f'(a)+(x-a)2f''(a)2!+(x-a)3f'''(a)3!+... Here…

Q: 4z-1 Find all Laurent series of f(z) = with center 1. 2+z-2

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Q: (8) Find the Taylor series for the function g(x) = 1/(x+3)2 at a = -2. Show your work.

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Q: (a) Find the Taylor series of cos 3x at x = 0. (b) Use the trigonometric identity cos'x = }(cos 3x +…

A: To Find: i. taylor series of cos3x at x=0.ii. use trignometric identitycos3x=14cos3x+3cosxto find…

Q: determine convergence or divergence using any method covered in the text so far.

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Q: find Fourier series of the Function: f (x) = x if (-n <x < n)

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Q: Determine whether the Fourier series of the following functions converge uniformly or not. A) f(x)…

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Q: 1. Find the trigonometric Fourier Series of the function defined by : filx)= x+# if (X...…... D -2…

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Q: The particular conditions that a function f(x) must fulfill in order that it may be expanded as a…

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Q: 8.8.12. Use a known Taylor series to conjecture the value of the limit. 2x

A: The Taylor series of e^-2x can be obtained from that of e^x  as shown on board.

Q: 0,π) Continue the f(x) function as an even or odd function and decompose it into a Fourier series.…

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Q: Find the Fourier series representation of the periodic function below          f(x) = ax2 + b  if…

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Q: i.Show the derivation  of Euler's formula for the solution of ODEs from the 1st principle utilizing…

A: (i) We can use Taylor series for derivation of Euler's formula. Taylor series for a function y=f(x)…

Q: 1. Sketch the even 2n - periodic extension and find the Fourier Cosine Series of the function…

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Q: 5. (a) Find the Fourier Series of the function (x) = 0 (x) = nx- x if 0< x< n. if -n < x < 0 (b) Use…

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Q: 5) a) Construct a Fourier Series for f(x) = (x2 12x)for 12 < x < 3n - b) Briefly explain the…

A: To give the Fourier series for  f(x)=x2 -12x for x∈0,3π We are considering this range since the x…

Q: Prove that f(x)={-3x if -pi<x<0, 3x if 0<x<pi is an even function using fourier series expansion.

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Q: AL IRAQIA UNIVERSITY Q3: 1. Find The Fourier series for the function defined by -n < x < 0] 0<x<n x…

A: Given function is fx=π+x       -π≤x≤0π-x          0≤x≤π The Fourier series of the function is…

Q: a. Let f(x) = sin(x). Find the Taylor polynomials up through order four of f cen- tered at x = .…

A: Note :As per our guidelines we are supposed to answer only one question if there are multiple…

Q: 1. Find the Fourier series of the function |-x if -4 <x< 0 f(x) S(x + 8) = f(x) if 0 <x< 4

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Q: 1. Find the Fourier Series coefficients representation of if -T < x < 0 f(x) TT – x if 0 < x< T…

A: Note:-  Fourier coefficients:- a0 =1L∫-LL fx dxa0 =1L∫-LL fx cosnπxL dxa0 =1L∫-LL fx sinnπxL dx…

Q: Find the Fourier series representation of the periodic function below          f(x) = ax2 + b  if…

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Q: Example 3) Expand f(x) = x - x²as a Fourier series in - 1<x<I.

A: The given problem is to find the Fourier series expansion of the given periodic function with given…

Q: 29-30 The limit of an indeterminate form as xXo can some- times be found by expanding the functions…

A:

Q: B/Find the Fourier series of the following: if 0 < x < 1/2 - 4 1/2 < x < 1 1<x < 3/2 3/2 < x < 2 f…

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Q: 3. Recall that a Fourier series will converge quickly if the periodic extension of the function is…

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.