Prove that if f is defined for | x | &lt; r and if there exists a constant B such that | f n ( x ) | ≤ Bfor all | x | &lt; r and n ∈ N , then the Taylor series expansion : ∑ n = 0 ∞ f ( n ) ( 0 ) x n n ! converges to f ( x ) for | x | &lt; r.

Question

Prove that if   f is defined for       |    x      |    &amp;lt;  r and if there exists a constant   B such that      |        f    n    (  x  )      |    ≤  Bfor all       |    x      |    &amp;lt;  r and   n  ∈      N  , then the Taylor series expansion :      ∑          n      =      0        ∞                      f                  (          n          )                    (      0      )              x        n                    n      !      converges to   f  (  x  ) for       |    x      |    &amp;lt;  r.

Cseszteq5j · Accepted Answer

A Taylor series converges if and only if its remainder term,       R    n    (  x  ) tends to zero. This is just a consequence from taylors theoremTheorem: If   f  (  x  )  =      T    n    (  x  )  +      R    n    (  x  ) , where       T    n   is the nth degree Taylor polynomial of f at a and       lim          n      →      ∞            R    n    (  x  )  =  0for       |    x  −  a      |    &amp;lt;  R , then f is equal to the sum of its taylor series on the interval       |    x  −  a      |    &amp;lt;  R That is because, as you suggested, we have      |        R    n    (  x  )      |    ≤  B            x              n        +        1                    (      n      +      1      )      !      andTaylors Inequality: If       |        f          n      +      1        (  x  )      |    ≤  M for       |    x  −  a      |    ≤  d , then the remainder,       R    n    (  x  ) of the Taylor series satisfies the inequality       |        R    n    (  x  )      |    ≤      M          (      n      +      1      )      !            |    x  −  a            |              n      +      1       for       |    x  −  a      |    ≤  dso now you must consider this,      lim          n      →      ∞        B            x              n        +        1                    (      n      +      1      )      !      and we that  B      lim          n      →      ∞                  x              n        +        1                    (      n      +      1      )      !        =  B  ∙  0  =  0thus we have that       |        R    n    (  x  )      |    ≤  0 i.e., the remainder will tend to zero as n approaches infinity , and the proof is finished.

Prove that if f is defined for |x|<r and if there exists a constant B such that |f^n(x)|≤B for all |x|<r and n in N, then the Taylor series expansion : sum_(n=0)^oo (f^(n)(0)x^n)/(n!) converges to f(x) for |x|<r.

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