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Lagrange Error

  • Lagrange error: difference between nth term partial sum and compete evaluation of series
  • Lagrange error bound:
  • Finding Lagrange error:
    • Differentiate function until derivative series pattern recognized
    • Apply to Lagrange formula for error bound

Lagrange Error

Find the Lagrange Error Remainder for appromixmating cosx with a MacLaurin series, with a constant c
Determine the MacLaurin Series to determine fn(x)cosx = ∑n = 0 [(( − 1)nx2n)/(2n)!] fn(x) = [(( − 1)nx2n)/(2n)!]Rn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [([(( − 1)n + 1x2(n + 1)(c)(x)(n + 2))/((2(n + 1) + 1)!)])/((n + 1)!)] = [(c( − 1)n + 1xn + 2(n + 1) + 2)/((n + 1)!(2(n + 1) + 1)!)]
Find the Lagrange Error Bound for appromixmating cosx with a MacLaurin series, with a constant c
Apply definition of Lagrange Error Bound
Rn(x) < max| [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] |Rn(x) < max| [(c( − 1)n + 1xn + 2(n + 1) + 2)/((n + 1)!(2(n + 1) + 1)!)] |
Find the Lagrange Error Remainder for appromixmating [1/(1 − x)] with a MacLaurin series, with a constant c
Determine the MacLaurin Series to determine fn(x)[1/(1 − x)] = ∑n = 0 (x)n fn(x) = (x)nRn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [((c)n + 1xn + 1)/((n + 1)!)]
Find the Lagrange Error Bound for appromixmating cosx with a MacLaurin series, with a constan c
Apply definition of Lagrange Error Bound
Rn(x) < max| [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] |Rn(x) < max| [((c)n + 1xn + 1)/((n + 1)!)] |
Find the Lagrange Error Remainder for appromixmating [1/(1 + x3)] with a MacLaurin series, with a constant c
Determine the MacLaurin Series to determine fn(x)[1/(1 + x3)] = ∑n = 0 ( − x3)n fn(x) = ( − x3)nRn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [(( − c3)n + 1xn + 1)/((n + 1)!)]
Find the Lagrange Error Bound for appromixmating cosx with a MacLaurin series, with a constant c
  • Apply definition of Lagrange Error Bound
Rn(x) < max| [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] |Rn(x) < max| [(( − c3)n + 1xn + 1)/((n + 1)!)] |
Find the Lagrange Error Remainder for appromixmating [1/(1 + 3x4)] with a MacLaurin series, with a constant c
Determine the MacLaurin Series to determine fn(x)[1/(1 + 3x4)] = ∑n = 0 ( − 3x4)n fn(x) = ( − 3x4)nRn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [(( − 3c4)n + 1xn + 1)/((n + 1)!)]
Find the Lagrange Error Bound for appromixmating [1/(1 + 3x4)] with a MacLaurin series, with a constant c
Apply definition of Lagrange Error Bound
Rn(x) < max| [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] |Rn(x) < max| [(( − 3c4)n + 1xn + 1)/((n + 1)!)] |
Find the Lagrange Error Remainder for appromixmating e − x with a MacLaurin series, with a constant c
Determine the MacLaurin Series to determine fn(x)e − x = ∑n = 0 [(( − 1)nxn)/n!] fn(x) = [(( − 1)nxn)/n!]Rn(x) = [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] = [([(( − 1)n + 1cn + 1)/(( n + 1 )!)]xn + 1)/((n + 1)!)] = [( − (1)n + 1cn + 1xn + 1)/(( (n + 1)! )2)]
Find the Lagrange Error Bound for appromixmating e − x with a MacLaurin series, with a constant c
  • Apply definition of Lagrange Error Bound
Rn(x) < max| [(f(n + 1)(c)(x − a)(n + 1))/((n + 1)!)] |Rn(x) < max| [( − (1)n + 1cn + 1xn + 1)/(( (n + 1)! )2)] |

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Lagrange Error

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Power Series: Lagrange Error 0:06
    • Lagrange Remainder
    • Lagrange Error Bound
  • Example 1 1:06
  • Example 2 3:27