Proving ∑ k = 1 ∞ ( S k ( x ) − f ( x ) ) Converges absolutley for | x | &lt; R

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Proving       ∑          k      =      1              ∞        (      S    k    (  x  )  −  f  (  x  )  ) Converges absolutley for       |    x      |    &amp;lt;  R

Michaela Alvarado · Accepted Answer

We have      ∑          k      =      0        ∞        ∑          n      =      k      +      1        ∞        |        a    n        x    n        |    =      ∑          n      =      0        ∞        ∑          k      =      0              n      −      1            |        a    n        x    n        |    =      ∑          n      =      0        ∞    n      |        a    n        x    n        |      .Now show that this series has the same radius of convergence as the original series       ∑          n      =      0        ∞        a    n        x    n  The initial rearrangement is allowed because all terms are non-negative, in the sense that the first and the second series are simultaneously convergent or divergent, and in the case of convergence their value is equal.

Proving <munderover> ∑ <mrow class="MJX-TeXAtom-ORD"> k =

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