Math 181 Calculus II

Power Series Convergence

JL

Convergence and Absolute Convergence

•

∑_*^∞C_N CONVerges iff

lim N → ∞  N ∑ *  Cn

exists (finite).

Compare to

•

∫_*^∞f(x)  dx CONVerges iff

lim X → ∞  ⌠ ⌡ X *  f(x)  dx

exists (finite).

•

∑_*^∞C_n CONVerges ABSsolutely iff ∑_*^∞|C_n| converges or

lim N → ∞  N ∑ *  |Cn|

exists (finite).

Compare to

•

∫_*^∞f(x)  dx CONVerges ABSsolutely iff ∫_*^∞|f(x)|  dx converges or

lim X → ∞  ⌠ ⌡ X *  |f(x)|  dx

exists (finite).

•

∑_*^∞|C_n| converges iff the sequence ∑_*^N |C_n| is bounded.

•

∫_*^∞|f(x)| converges iff the function ∫_*^X |f(x)|  dx is bounded (as X → ∞).

Comparison Test for Absolute Convergence

•

If

0 ≤ An ≤ Bn,

then

0 ≤ ∞ ∑ *  An ≤ ∞ ∑ *  Bn.

So that if the bigger series ∑_*^∞B_n CONVerges, the smaller series ∑_*^∞A_n CONVerges also.

If the smaller series ∑_*^∞A_n DIVerges, the bigger series ∑_*^∞A_n DIVerges also.

•

If

0 ≤ f(x) ≤ g(x),

then

0 ≤ ⌠ ⌡ ∞ *  f(x)  dx ≤ ⌠ ⌡ ∞ *  g(x)  dx.

So that if the bigger integral ∫_*^∞g(x)  dx CONVerges, the smaller integral ∫_*^∞f(x)  dx CONVerges also.

If the smaller integral ∫_*^∞f(x)  dx DIVerges, the bigger integral ∫_*^∞g(x)  dx DIVerges also.

Ratio Test for ABSolute CONVergence

For the series ∑_*^∞C_N, suppose that

lim n → ∞  ⎢ ⎢ Cn+1 Cn ⎢ ⎢ = L.

•

If 0 ≤ L < 1, the series ∑_*^∞C_N CONVerges ABSolutely.

•

If 1 < L ≤ ∞, the series ∑_*^∞C_N DIVerges.

•

If L = 1, we are not sure - additional information is needed about DIVergence or CONVergence and/or ABS0lute CONVergence.

Power Series, Radius of Convergence, and Interval of Convergence

•

For a power series ∑_n=0^∞ a_n xⁿ, there is a number R, 0 ≤ R ≤ ∞ for which

∞ ∑ n=0  an xn ⎧ ⎪ ⎨ ⎪ ⎩ CONVerges ABSolutely for |x| < R, DIVerges for |x| > R.

The number R is called the radius of convergence of the power series. R can often be determined by the Ratio Test.

•

If the power series ∑_n=0^∞ a_n xⁿ, converges for x = x₀, then for all x, |x| < |x₀| the power series CONVerges ABSolutely. Thus the radius of convergence , R,is greater than or equal |x₀|.

•

If f(x) is represented by a convergent power series for |x| < R, then for |x| < R, its derivative is represented by the convergent series ∑_n=1^∞ n a_n xⁿ⁻¹:

If

f(x) = ∞ ∑ n=0  an xn, |x| < R,

then

f′(x) = ∞ ∑ n=1  n an xn−1 = ∞ ∑ n=0  (n+1) an+1 xn, |x| < R,

and

⌠ ⌡ x 0  f(t)  dt = ∞ ∑ n=0  an n+1 xn+1 = ∞ ∑ n=1  an−1 n xn, |x| < R

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On 22 Feb 2012, 13:24.