Math 181 Calculus II

Power Series

JL

Viewpoints

When we write

f(x) = a0 + a1 x + a2 x2 + …an xn + ..., = ∞ ∑ n=0  an xn, an = f(n)(0) n! PN(x) = PN,0(x) = a0 + a1 x + a2 x2 + …aN xN, = N ∑ n=0  an xn, RN(x) ≡ f(x) − PN(x),

there are two contexts in which we are saying

f(x) ≈ PN(x).

The local context is that, fixing N,

f(x) = PN(x) + error,

where, as x → 0 the error [ = R_N(x)] is much much smaller than x^N - the last and smallest term which has been included in P_N.

Thus, fixing N, P_N(x) is a very good approximation for x near 0.

The global context is that for all x in an interval,

f(x) = PN(x) + error,

where, as N → ∞, the error [ = R_N(x)] is small - the infinite series ∑_n=0^∞ a_n xⁿ converges to the value f(x).

Thus, fixing x [or varying x over a fixed interval], P_N(x) is a very good approximation for N very large.

From the local viewpoint, since

P0(x) = f(0),

the statement f(x) ≈ P₀(x) gives the statement f(x) ≈ f(0) for x near 0 - or f(x) is continuous at x=0.

Note that

P1(x) = f(0) + f′(0)·(x − 0)

gives the tangent line approximation to f(x).

The global viewpoint is required to give an efficient way to calculate, e. g., e^x, for, say −0.1 ≤ x ≤ 0.1

For example, in treating continuous compounding of interest, we might need e^r, with r=6.1% = .061 or r=6.9% = .069. A very good approximation is

er ≈ 1 + r + r2 2 , −0.1 ≤ x ≤ 0.1.

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On 08 Feb 2012, 18:33.