Prove convergence of the implicit midpoint method by following the steps below. a) Use the order...

 

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Prove convergence of the implicit midpoint method by following the steps below. a) Use the order of the Implicit Midpoint Rule and the Lipschitz continuity of f(t, y) to show that eitie+hX/2(e;+e+1) +0(h), i = 0, 1,... b) Using the definition of O(h) and part a), show that 1+ h/2 C -e; + 1-h/2 1 - hA/2 h, i = 0, 1,... c) Prove by induction that eit1 < +h/21 = ( ( 1 + X/2 ) - 1) 1- h/2, h, i = 0,1,... d) Using the results above conclude that the implicit midpoint method converges. Prove convergence of the implicit midpoint method by following the steps below. a) Use the order of the Implicit Midpoint Rule and the Lipschitz continuity of f(t, y) to show that eitie+hX/2(e;+e+1) +0(h), i = 0, 1,... b) Using the definition of O(h) and part a), show that 1+ h/2 C -e; + 1-h/2 1 - hA/2 h, i = 0, 1,... c) Prove by induction that eit1 < +h/21 = ( ( 1 + X/2 ) - 1) 1- h/2, h, i = 0,1,... d) Using the results above conclude that the implicit midpoint method converges.