Simplification of In x + In(x + 1) = 0 is O x2 + x-1 = 0; x > 0 O x? + x + 1 = 0; x > 0 O x2 + x - e = 0; x > 0 O x² + x + e = 0; x > 0

Q: Express the integral as an infinite sum. | In (1 + i') dt, x| < 1 Choose the infinite sum that…

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Q: 8). The series x 0 equals 2² (1!) 2* (2!)? 2° (3!)? (i) 'v½ (x) (ii) jo(x) (iii) x Jo (x) (iv) x J/2…

A: 8) xj0(x) Option ( iii)     j0(x) =1-x222+x42242-x6224262+...xj0(x) =x-x322+x52242-x7224262+....…

Q: 6. Write the Riemann sum as an integral.

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A: We have given that the theorem as Assume that f1x is Og1x and f2x is Og2x. Then , (i) f1+f2x is Ogx…

Q: 23. Using mathematical induction, show that dn n [F (x) · g(x)] = D dk [f(x}] • n dn-k dxn k) dxk…

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Q: Expand cos'x as a series of ascending powers of x.

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Q: =1 using Cauchy's Residue theorem.

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Q: limit as n approaches +infinity of (n/(e)^n)

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Q: 8 Let k(x) = |x|", n = 1,2, ... Is k differentiable for all x and every n? Find a formula for k' (x)…

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Q: a) In x-3/In x +2=0 giving answer in exact form and decimal approximation to 3 decimal places

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Q: 15. Expand. 1+x 21 obtain an 100 in ascending powers of x up to the term in x. By substituting x =…

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Q: ) How alutiino a. many ths to the eguotin X, + X2 + X3 +X4 + X5 = 10, uhere poitine integoro ? X,,…

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Q: We consider the integral Evaluate the approximative value of T by Trapeze method then Simpson's…

A: Solution:Given   T=∫141xdxEvaluate the approximate value of T:1. Trapeze Method2. Simpson's…

Q: Using mathematical induction, show that for n&gt;1 dn /dxn ln x = (-1)n-1 (n - 1)!/xn

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Q: 2COS X-I Sin x, Yna1 = 2cosxy, – Yn-1+ n21 and 0<xs. (i) Use mathematical induction to prove that…

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Q: Show that tan(x) > x for 0 0 for 0 1 for 0 x for on 0 < x < -

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

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Q: 5. Evaluate z z = tanh(4-j5)

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Q: Let (r.) be a sequence of positive numbers with Intl = L. lim n0 In Let 0 <e< L. Show that there…

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Q: Use series to evaluate the limit. 4 3 sin(2x) – 2x + lim 4/3

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Q: 1) In order to use the integral test, we must verify that the sequence is decreasing (and positive…

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Q: 4. i) Prove that the series sin nx n2 n=1 converges pointwise on R, and uniformly on any finite…

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Q: 26) ) Show that the reductión formula for 1ー2 Sec" ndx = Sec atan n t n-2 n-- n-1 ) Hence use tta…

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Q: What is the interval of convergence of Taylor Polynomial of f(x)=cos x at a=pi/3

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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: Express f(x) = n + x_for -<x<n as a Fourier series.

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Q: Define a power series  f(x)=Σn≥0(anxn)=x+x2+2x3+3x4+5x5+8x6.

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Q: 2°) Use integration by parts to prove that, for any positive integer n, (2n + 1) In = 2n In-1. 3°)…

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Q: Prove without using induction that  Fn+1^2 - Fn^2 = Fn-1 * F n+2 where F represents a Fibonacci…

A: To prove, Fn+12-Fn2=Fn-1×Fn+2

Q: 21. Write the equation 1+3+6+10+…+ п(n+1) 2 п(п+1)(п+2) 6. using summation notation. Prove that the…

A: The equation 1+3+6+10+...+n(n+1)2=n(n+1)(n+2)6 is written as: ∑k=1nk(k+1)2=n(n+1)(n+2)6So the…

Q: Prove that 1+ x In 2 (G tanh- x = - 1< x < 1. | 1 - x

A: To prove,  tanh-1x=12ln1+x1-x

Q: EXAMPLE Find Fourier series for - f(x) = { ,0 sxSn In + x -nsxs0 (n- x

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Q: 4. Use the integral expression you wrote in #3 and the Fundamental Theorem of Calculus to find the…

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Q: 4. The gamma function, which plays an important role in advanced applications, is defined for n >1…

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Q: 5. Prove that . Hint: Compute e-' from the continued frac e - 1 1 + 2 + 3 + ·…. tion for et given in…

A: When a series is in the form ∑k=1∞-1k-1α1α2α3...αk, we can express it as a continued fraction as…

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: 12 Σ 1 – x2 where 0 <x < 1. n=0

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Q: lim (In æ – !) x→0+ O In (-∞) O In (0) 1 -1 O In (0)

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Q: ries of f(x) = lor series. =0 =0 =0 (−1)²+1 (-1)²+¹ (-1)". 1 (x-7)" 8n+1 (x (x − 8)¹+1 7" E(-1)", =0…

A: We have to find the Taylor series from the given condition and also indentify  the valid interval

Q: 1.Find the interval and radius of conv. of In(1 - x2) ?

A: Maclurin series representation for ln(1- x2 ) is  ∑n=0∞-(x)2n +2n+1   (Why?) Explanation As ln(1+x)…

Q: 2. Determine the positive real root of the function f(x) = In(x²) – 2 with initial guess [5/2, 3]…

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Q: 5. Expand the following function in Fourier series where f(x) = { show that f(x): = + 22-1 [(-1)^-1…

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Q: 13 By use of De Moivre's Theorem, evaluate (-+i) .

A: We have to evaluate -12+32i13 by using De Mover's Theorem.

Q: Working directly from definition, prove that if zn and w, are se- quences of complex numbers with…

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Q: Define the sequence of functions {fn(x)} on [0, 1] by { 2n2x; 2n(1 — пх); т. fn(x) : 0; n Find f(x)…

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