Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...
Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...
Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...
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The Pro<strong>of</strong><br />
Combining all these ideas in just the right way yields<br />
( ∫ )<br />
2n + 1 t=h<br />
lim<br />
f(t)P n (t/h)dt P n (z/h) = f(n) (0)<br />
h→0 + 2h<br />
n!<br />
for each n ≥ 0.<br />
t=−h<br />
<strong>Taylor</strong> <strong>Series</strong> <strong>are</strong><strong>Limits</strong> <strong>of</strong> <strong>Legendre</strong> <strong>Expansions</strong> – p.11/15