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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.