# Series solutions near a regular singular point

## Main Question or Discussion Point

For solving a series solution near a regular singular point with the Frobenius method, why is it that the indices of summation derivatives aren't shifted?

For example, in my textbook and lecture notes

y = $$\sum$$A$$_{}n$$x$$^{}n+r$$ from n=0 to infinity

y' = $$\sum$$(n+r)A$$_{}n$$x$$^{}n+r-1$$ from n=0 to infinity

y'' = $$\sum$$(n+r)(n+r-1)A$$_{}n$$x$$^{}n+r-2$$ from n=0 to infinity

But shouldn't the index for y' be from n=1 to infinity because it shifts up when you take the derivative of a summation? Shouldn't the index for y'' be from n=2 to infinity?

Thanks.

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tiny-tim
Homework Helper
Welcome to PF!

Hi phrankle! Welcome to PF!

Of course, you could change it to ∑A(n+1)x^(n+r), and do it from n = -1 to ∞ …

but that would be unnecessarily complicated, and you could easily make a mistake …