What is Lagrange Error and how do you find the value for it \[M\] ?

Hint: We need to know the Taylor series equation in terms of \[f\left( x \right)\] around \[x = a\] . In that formula next, we put \[k = n\] . We need to know the basic equation for \[{P_n}\left( x \right)\] and \[{R_n}\left( x \right)\] . By using these equations next we would relate the Taylor series equation with the equations of \[{P_n}\left( x \right)\] and \[{R_n}\left( x \right)\] . Also, we need to substitute the second theorem of the mean in the equation \[{R_n}\left( x \right)\] to find the term \[M\] .

Complete step by step solution:
Consider the Taylor series of a function \[f\left( x \right)\] around \[x = a\] ,
 \[f\left( x \right) = \sum\limits_{k = 0}^\infty {\dfrac{{{f^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} \]
If we stop the Taylor series at \[k = n\] we have,
 \[f\left( x \right) = {P_n}\left( x \right) + {R_n}\left( x \right)\]
Where,
 \[{P_n}\left( x \right) = \sum\limits_{k = 0}^\infty {\dfrac{{{f^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} \]
And it can be demonstrated that rest can be expressed as,
 \[{R_n}x = \dfrac{1}{{n!}}\int\limits_n^x {{f^{\left( {n + 1} \right)}}} \left( t \right){\left( {x - t} \right)^n}dt\]
Applying the second theorem of the mean to this integral we have,
 \[{R_n}\left( x \right) = \dfrac{1}{{\left( {n + 1!} \right)}}{f^{\left( {n + 1} \right)}}\left( \xi \right){\left( {x - a} \right)^{n + 1}}\]
Where \[\xi \] is a point between \[x\] and \[a\]
Clearly if in the interval delimited by \[x\] and \[a\] we have,
 \[\left| {{f^{\left( {n + 1} \right)}}\left( \xi \right)} \right| \prec M\]
Then
 \[\left| {{R_n}\left( x \right)} \right| \leqslant \dfrac{M}{{\left( {n + 1} \right)!}}{\left| {x - a} \right|^{n + 1}}\]
So, the correct answer is “ \[\left| {{R_n}\left( x \right)} \right| \leqslant \dfrac{M}{{\left( {n + 1} \right)!}}{\left| {x - a} \right|^{n + 1}}\] ”.

Note: Remember the second theorem of the mean to solve these types of questions. Note that \[\xi \] is a point between \[x\] and \[a\] . Also, note that if \[\left| {{f^{\left( {n + 1} \right)}}\left( \xi \right)} \right|\] is less than \[M\] then \[\left| {{R_n}\left( x \right)} \right|\] is also less than or equal to \[\dfrac{M}{{\left( {n + 1} \right)!}}{\left| {x - a} \right|^{n + 1}}\] . Also, note that \[f\left( x \right)\] is also can be written as the sum of \[{P_n}\left( x \right)\] and \[{R_n}\left( x \right)\] .