Differential-Equations Elementary Power Series Solutions, guaranteed and verified 100% Pass

1


Elementary Power Series Solutions


A power series around 0 is of the form:
∑∞ 𝑛 2 𝑛
𝑛=0 𝑐𝑛 𝑥 = 𝑐0 + 𝑐1 𝑥 + 𝑐2 𝑥 + ⋯ + 𝑐𝑛 𝑥 + ⋯
A power series around 𝑎 is of the form:
∑∞ 𝑛 𝑛
𝑛=0 𝑐𝑛 (𝑥 − 𝑎) = 𝑐0 + 𝑐1 (𝑥 − 𝑎) + ⋯ + 𝑐𝑛 (𝑥 − 𝑎) + ⋯



𝑥𝑛 𝑥2 𝑥3 𝑥𝑛
Ex. 𝑒 𝑥 = ∑∞
𝑛=0 =1+𝑥+ + +⋯ +⋯
𝑛! 2! 3! 𝑛!
(−1)𝑛 𝑥 2𝑛 𝑥2 𝑥4 (−1)𝑛 𝑥 2𝑛
cos 𝑥 = ∑∞
𝑛=0 =1 − + + ⋯+ (2𝑛)!
+⋯
(2𝑛)! 2! 4!

(−1)𝑛 𝑥 2𝑛+1 𝑥3 𝑥5 (−1)𝑛 𝑥 2𝑛+1
sin 𝑥 = ∑∞
𝑛=0 =𝑥 − + +⋯+ (2𝑛+1)!
+⋯
(2𝑛+1)! 3! 5!
1
= ∑∞ 𝑛 2 3 𝑛
𝑛=0 𝑥 = 1 + 𝑥 + 𝑥 + 𝑥 + ⋯ + 𝑥 + ⋯
1−𝑥
1
= ∑∞ 𝑛 𝑛 2 3 𝑛 𝑛
𝑛=0(−1) 𝑥 = 1 − 𝑥 + 𝑥 − 𝑥 + ⋯ (−1) 𝑥 + ⋯
1+𝑥



Notice that means:
(2𝑥)𝑛 (2𝑥)2 (2𝑥)3
𝑒 2𝑥 = ∑∞
𝑛=0 = 1 + 2𝑥 + + +⋯
𝑛! 2! 3!
(−1)𝑛 (3𝑥)2𝑛 (3𝑥)2 (3𝑥)4
cos 3𝑥 = ∑∞
𝑛=0 =1− + +⋯
(2𝑛)! 2! 4!



Def. If the Taylor Series of 𝑓(𝑥) converges to 𝑓 (𝑥 ) for some open interval
containing 𝑥 = 𝑎, we say 𝑓 is analytic at 𝑥 = 𝑎.

, 2


Ex. 𝑓(𝑥 ) = 𝑒 𝑥 is analytic everywhere.
1
𝑓 (𝑥 ) = is analytic everywhere except 𝑥 = 1.
1−𝑥

All polynomials and rational functions whose denominators are not 0 are
analytic.




Power Series Operations
Power series operations are similar to those of polynomials.

If 𝑓 (𝑥 ) = ∑∞ 𝑛
𝑛=0 𝑎𝑛 𝑥 , 𝑔(𝑥 ) = ∑∞
𝑛=0 𝑏𝑛 𝑥
𝑛

then,

𝑓(𝑥 ) ± 𝑔(𝑥 ) = ∑∞
𝑛=0(𝑎𝑛 ± 𝑏𝑛 )𝑥
𝑛

and
𝑓(𝑥 )𝑔(𝑥 ) = (𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + ⋯ )(𝑏0 + 𝑏1 𝑥 + 𝑏2 𝑥 2 + ⋯ )
= 𝑎0 𝑏0 + (𝑎0 𝑏1 + 𝑎1 𝑏0 )𝑥 + (𝑎0 𝑏2 + 𝑎1 𝑏1 + 𝑎2 𝑏0 )𝑥 2 + ⋯.


Given a power series, ∑∞ 𝑛
𝑛=0 𝑐𝑛 𝑥 , we often want to know for what values of 𝑥
the series converges.