Prove the following integral. ., xPn-1(x)P,(x)dx = 2n (2n-1)(2+1) 2 use (n + 1)Pn+1(x) – (2n +1)xPn(x) + nPn-1(x) = 0, L, Pn(x)Pm (x) dx = 0, S, Pn(x)2 dx = 2n+1
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of degree 0 and 1, 3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of...
7n Use Mathematical Induction to prove that Σ 2-2n+1-2, for all n e N
Prove: without using l'hopital's rule. infinity 2n-1 ln(2) (2n-1) n infinity 2n-1 ln(2) (2n-1) n
1. Let 21,...,m ER be m distinct real numbers. Define m (p, q)m = p(x;) g(x3), j=1 for all p, q E P = {real polynomials}. Does (-;-)m define an inner product on P? If so, then prove it. If not, then give a counterexample. For which n e N does (-:-)m define an inner product on Pn = {p € P: deg p <n}. Make sure to justify your answer fully!
Find a Maclaurin series for f(x). (Use (2n)! —for 1:3:5... (2n – 3).) 2"n!(2n-1) X Rx) = (* V1 +48 dt . -*** * 3 n = 2 Need Help? Read It Talk to a Tutor
2n 3. Prove that lim n+on+ 1 2.
Pt 1 pt 2 pt 3 pt 4 Please Answer every question and SHOW WORK! Determine whether the series n-1 Σ (2n)! 2". (2n! converge or diverge 1. both series converge 2. only series II converges 3. only series I converg es 4. both series diverge Determine whether the series 2! 1515.9 1-5.9-13 3! 4! 7m 1.5.9..(4n -3) is absolutely convergent, conditionally con- vergent, or divergent 1. conditionally convergent 2. absolutely convergent 3. divergent Determine which, if any, of the...
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
tan-i (-1+-- 2. Express the limit lini Σ 2n 2 as a definite n 2n integral. Make sure to fully justify your work. Hint: What is equal to? tan-i (-1+-- 2. Express the limit lini Σ 2n 2 as a definite n 2n integral. Make sure to fully justify your work. Hint: What is equal to?