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A very simple question: How can I tell to Mathematica that: $\begin{equation} x*\sum_{k=0}^{\infty}\,b_kx^k=\sum_{k=0}^{\infty}\,b_kx^{k+1}\end{equation} $

I tried to multiply but Mathematica gives me only $\begin{equation} x*\sum_{k=0}^{\infty}\,b_kx^k\end{equation}. $ I am trying to apply Frobenius method, and I need to collect the same powers of $x$ in my final equation.

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You can use ReplaceAll:

sum = Sum[Subscript[b, k] x^k, {k, 0, Infinity}];
x sum  /. a_. Sum[s_, r__] :> Sum[a s, r]
TeXForm @ %

$\sum _{k=0}^{\infty } b_k x^{k+1}$

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