Show that f"(0) = 0. Use induction and the ideas from the above to show that f(k)(0) = 0 for every k and conclude that f = 0. %3D Assume that g(r) =anr" on (–R, R). Instead of assuming that f(In) = 0 assume n=0 that f(x,) = g(rn) for all n. Show that f(x) = g(x) on (–R, R).

Q: dt converges. There is no value of r such that (3t)-" 14.

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Q: ind the hal f CoSine range Seriesfor in the Intervat Hence that the function (X-H 32 1=1 (2n-1)2 12…

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Q: 1+2x Expand ()* in ascending powers of x up to and including the term in x³. B, 1 assuming x =…

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A: Given - The function f(x) has its fifth derivative continuous on (−∞,∞). If |f(5)| &lt; 6 on the…

Q: Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0<x s -n² (x…

A: Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

Q: Suppose {fn}=1 and {g„}=1 converges uniformly on E where {fn}=1 and {gn}n=1 are sequences of bounded…

A: Let fnn=1∞ is converges uniformly to f and gnn=1∞ is converges uniformly to g on E and fnn=1∞ and…

Q: Let Zn be the size of the nth generation in a GW process with Zo = 1, E(Z¡) = µ. Show that E(Z„Zm) =…

A:

Q: Find the constant term in the expansion

A: We have to solve the problem

Q: Find an example of fn → f in measure, but for every x, fn(x) does not converge to f(x).

A: We have to find an example of fn→f in measure, but for every x, fn(x) does not converge to f(x).

Q: 2. Show that X H→ In x is increasing on (0,0) and that In 1 = 0:

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Q: 3, Re call Eudlid's proof of In finitude * primes! Shouw that T Cx): = 1{ ps X: p-pinae }12 logz…

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Q: The approximation of the root x' of the funetion f(x) = x* + x* – 2x² – 3x – 3 accurate to within…

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Q: Prove that the seugences or series of functions converges pointwise but not uniformly on [0,1] 1.…

A: We have to show that the following sequence or series of functions converges pointwise but not…

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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Q: Suppose that F(s) = ℒ{f(t)} exists for s &gt; a ≥ 0. Show that if k is a positive constant, then…

A:

Q: Suppose that 0< fn < In on X for all n e N. If gn 30 on X, then fn 30 on X.

A: Solution:-

Q: 1 d.x x In (ex)

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Q: sin(²) En=1 is 18 √n a. 1 b. 12/12 c. Converge d. Diverge d O C O a O b O

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Q: (i) Find the constant term in the expansion of x+-

A: You have given 3 different questions in picture, without mentioning any particular questions. We can…

Q: 11. (a) Let F(s) = L{f(t)}, where f(t) is piecewise continuous and of exponential order on [0, ∞o).…

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Q: Calculate the value of f(1,05) from the function f(x)=3/x with accuracy of two decimal place, by…

A: Given: The function f(x)=3x and x0=1

Q: CoS“ n In n а. n2 n³ n=1

A: Solution: a) ∑n=1∞cos2(n)n2+ln (n)n3

Q: 1. For r < 1, define fn(x) = ²x. (a) Prove that for x < 1, k=1 ƒ(x) = lim_ƒn(x) = 84x x(1+x) (1-x)³…

A:

Q: Let pn be the Taylor polynomial of degree n for f(x) = log(1 – x) about a = 0. How large should n be…

A: Let pn be the Taylor polynomial of degree n for fx=log1−x about a=0. The Taylor polynomial is given…

Q: Let n >1. Suppose that f has (n – 1)-th order continuous derivative and is n-times differentiable in…

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Q: k+ln k > ak, where aj = and ak+1 ak k+10 %3D k=1

A:

Q: Suppose f(x) > 0 and g(x) > 0 and that | f(x)dx diverges. If f(x) > g(x), what can we conclude about…

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Q: Suppose X - Bernoulli (0.5) and Y ~ Exponential (2). Suppose further that X and Y are independent.…

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Q: 4. Show that the Taylor polynomials for f(x) = In(1 + x) can be used to approximate f(s) for s e…

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Q: Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0,…

A:

Q: suppose the quantity of a substance, Q, decreases by 6.5% in 10 hours 1. find a formula for the…

A: The decay model is represented by,

Q: Let pn be the Taylor polynomial of degree n for f(x) = log(1 – x) about a = 0. How large should n be…

A:

Q: Show that -E{t, log(sigma(w*x)) + (1 – t;) log(1 – sigma(w"x;))} = E, log(1+ e-yw* x %3D Where…

A:

Q: 1. f(x) da converges, then If f(x) ≤ g(x) for all z in [a, +∞o) and converges. g(x) dx also

A: Given

Q: 4. Exercise §3.4 #47. Let n €Z with n> 0. Prove that E - a(n)

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Q: The error function, erf(x) = Te -* dt, important in probability and in the theories of heat flow and…

A: Simpson's rule: For a continuous function fx, an even number n, and △x=b-an, the following is true.…

Q: (a) Show that v.00 -p 0, where v++ denote the Laplacian operator (b) Using Taylor series method to…

A: The dot product of the vectors a and b is defined as…

Q: For n e N, define fn : [-1, 1] → [0, 1] by fn(x) = (1 – x2)". For all 8 e (0,1), define Is = [–1,…

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Q: Suppose {fn}n=1 converges uniformly on E. Show {fn}n=1 converges pointwise on E.

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Q: If fn : R → [0, 0) is a sequence of integrable functions, fn + f a.e. as n → 0, and S fn + 0. Prove…

A:

Q: The generating function w (x, t) = (1– 2xt + t2)-1 is given. (a) Show that it satisfies the identity…

A: Note: Since problem 7 is not given which is needed to solve part b, we will solve part a. Given:…

Q: dx heorem 1.7.1 is convergent if and only if µ < 1. (x – a)H a

A:

Q: that converges to a function on R ii)show n х 1=0n!

A: For this, we will use the Ratio Test, which can be written for a series ∑n=0∞an as: limn→∞ an+1 an…

Q: 5. Show that |Logz|<|In(z)|+ t, for all z # 0.

A:

Q: 3. Use alternating test En=1 (-1)" _2n² 2 n¯ +1 4. Use integral test, identify what is the value…

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Q: 1 Px(x) = x(x+1) 0, x = 1,2,..., otherwise. F(x) x 1. An alternative method of computing E[X] on…

A: *Answer:

Q: (B) Let {fn(x)}1 = {r ln x. cos(x – 1)"} be a sequence of meaurable functions defined over [1,2].…

A:

Q: Suppose that fn(x) and f(x) are functions R –→ R. Suppose also that fn converges uniformly to f as n…

A: Suppose that fnx and fx are functions from ℝ to ℝ such that fn converges uniformly to f as n→∞.  To…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.