{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

06OperationsPowerSeriesAMS

06OperationsPowerSeriesAMS - Operations on Power Series...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Operations on Power Series Let’s start with 2 power series about x0 : ∞ f ( x) = n=0 ∞ a n ( x − x0 ) n = a 0 + a 1 ( x − x0 ) 1 + a 2 ( x − x0 ) 2 + a 3 ( x − x0 ) 3 + . . . b n ( x − x0 ) n = b 0 + b 1 ( x − x0 ) 1 + b 2 ( x − x0 ) 2 + b 3 ( x − x0 ) 3 + . . . n=0 g ( x) = each of which converge absolutely for x ∈ (x0 − R , x0 + R). c ∈ R = (−∞, +∞) , x ∈ ( x0 − R , x 0 + R ) , def Let: m ∈ N = {1, 2, 3, . . . } β ∈ ( x0 − R , x0 + R ) def α ∈ ( x 0 − R , x0 + R ) , Then (note we excluded the endpoints of (x0 − R , x0 + R), ie. we excluded x = x0 ± R since things sometimes don’t hold at the endpoints): ∞ f ( x) + g ( x) = (∗) n=0 ∞ ( a n + b n ) ( x − x0 ) n ( a n − b n ) ( x − x0 ) n n=0 ∞ f ( x) − g ( x) = (∗) c f ( x) = ( x − x0 ) ∞ m (∗) n=0 ∞ c a n ( x − x0 ) n ∞ f ( x) = (∗) n=0 ∞ a n ( x − x0 ) ( x − x0 ) Dx (an (x − x0 )n ) = n=0 ∞ m n = n=0 ∞ an (x − x0 )m+n Dx [f (x)] = Dx n=0 a n ( x − x0 ) n So (∗) = nan (x − x0 )n−1 n=0 Dx [f (x)] = n=1 ∞ nan (x − x0 )n−1 β β β ∞ f (t) dt = α α n=0 a n ( t − x0 ) n β dt = (∗) n=0 ∞ (an (t − x0 )n ) dt = α an (t − x0 )n+1 n+1 n=0 ∞ t=β t=α So α f (t) dt = an (β − x0 )n+1 − (α − x0 )n+1 n+1 n=0 Furthermore, f (x) · g (x) is what you think it should be for x ∈ (−R , R). If b0 = 0, (x) then f (x) is what you think it should be but for only x suﬃciently small enough. g ...
View Full Document

{[ snackBarMessage ]}