series and exp

1. Binomial theorem
Binomial is a Polynomial that has two terms
(x + y)0 = 1
(x + y)1 = (x + y)
(x + y)2 = x2 + 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
How do we get general terms like (x + y)n and n is large
(𝑥 + 𝑦)𝑛
= 0
𝑛
𝐶𝑥𝑛
+ 1
𝑛
𝐶𝑥𝑛−1
𝑦 + 2
𝑛
𝐶𝑥𝑛−2
𝑦2
+ ⋯ + 𝑟
𝑛
𝐶𝑥𝑛−𝑟
𝑦𝑟
+ ⋯ + 𝑛
𝑛
𝐶𝑦𝑛
The coefficient of rth term 𝑟
𝑛
𝐶 =
𝑛!
𝑛−𝑟 ! 𝑟!
=
𝑛
𝑟
(𝑥 + 𝑦)𝑛
= 𝑥𝑛
+ 𝑛𝑥𝑛−1
𝑦 +
𝑛(𝑛 − 1)
2!
𝑥𝑛−2
𝑦2
+ ⋯ 𝑟
𝑛
𝐶 𝑥𝑛−𝑟
𝑦𝑟
+ ⋯ + 𝑦𝑛
General expression
Finite series, if n is positive integer

2. Binomial Series
If we put
x = 1
y = x
n = p (and we allow p as negative or fractional )
The series becomes (1 + 𝑥)𝑝
.
The series is known as Binomial series
(1 + 𝑥)𝑝
=
𝑛=0
∞
𝑛
𝑝
𝐶 𝑥𝑛
= 1 + 𝑝𝑥 +
𝑝(𝑝 − 1)
2!
𝑥2
+
𝑝(𝑝 − 1)(𝑝 − 2)
3!
𝑥3
+ ⋯
𝑟
𝑛
𝐶 =
𝑛!
𝑛 − 𝑟 ! 𝑟!
=
𝑛
𝑟
(𝑥 + 𝑦)𝑛
= 𝑥𝑛
+ 𝑛𝑥𝑛−1
𝑦 +
𝑛(𝑛 − 1)
2!
𝑥𝑛−2
𝑦2
+ ⋯ 𝑟
𝑛
𝐶 𝑥𝑛−𝑟
𝑦𝑟
+ ⋯ + 𝑦𝑛
The infinite series converges if |x|<1

3. Example 1
(1 + 𝑥)𝑝
=
𝑛=0
∞
𝑛
𝑝
𝐶 𝑥𝑛
= 1 + 𝑝𝑥 +
𝑝(𝑝 − 1)
2!
𝑥2
+
𝑝(𝑝 − 1)(𝑝 − 2)
3!
𝑥3
+ ⋯
say p = -1
1
1 + 𝑥
= 1 − 𝑥 +
−1(−2)
2!
𝑥2
+
−1(−2)(−3)
3!
𝑥3
+ ⋯
= 1 − 𝑥 + 𝑥2
− 𝑥3
+ ⋯ =
𝑛=0
∞
(−𝑥)𝑛
say x = 0.01 1
1 + 𝑥
≈ 1 − 𝑥
1/1.01=0.990099…
1-0.01=0.99

4. Example 2
𝑓(𝑥) = 1 + 𝑥
(1 + 𝑥)𝑝
=
𝑛=0
∞
𝑛
𝑝
𝐶 𝑥𝑛
= 1 + 𝑝𝑥 +
𝑝(𝑝 − 1)
2!
𝑥2
+
𝑝(𝑝 − 1)(𝑝 − 2)
3!
𝑥3
+ ⋯
𝑓 𝑥 = 1 +
1
2
𝑥 +
1/2(−1/2)
2!
𝑥2
+
1/2(−1/2)(−3/2)
3!
𝑥3
+ ⋯
= 1 +
1
2
𝑥 −
1
8
𝑥2
+
1
16
𝑥3
− ⋯

5. Fourier Series
sinx =
(-1)n
x2n+1
(2n+1)!
n=0
¥
å = x -
x3
3!
+
x5
5!
-
x7
7!
-×××
Taylor series: An oscillating series expressed as infinite polynomial series
Recap

6. Fourier Series
1
-1
0 0.5 1
Length
Temp
HOT
COLD

7. Fourier Series 1
-1
0 0.5 1
Length
Temp
HOT
COLD
𝑇 =
4
𝜋
cos(1𝜋𝑥)
1
−
cos 3𝜋𝑥
3
+
cos 5𝜋𝑥
5
−
cos 7𝜋𝑥
7
+ ⋯

8. Fourier Series
𝑇 =
4
𝜋
cos(1𝜋𝑥)
1
−
cos 3𝜋𝑥
3
+
cos 5𝜋𝑥
5
−
cos 7𝜋𝑥
7
+ ⋯ =
1 𝑖𝑓 𝑥 < 0.5
0 𝑖𝑓 𝑥 = 0
−1 𝑖𝑓 𝑥 > 0.5
1
-1
0 0.5 1
Length
Temp
HOT
COLD

9. Fourier Series
2.Taylor series can give a good local approximation (given you are within the radius of convergence);
Fourier series give good global approximations
1. Many phenomena in nature repeat themselves (e.g., heartbeat, songbird singing)
 Might make sense to ‘approximate them by periodic functions’
3. Fourier series gives us a means to transform from the time domain to frequency domain and vice
versa (e.g., via the FFT)

10. time waveform recorded from ear canal
... zoomed in
Fourier transform
Time Domain Spectral Domain

11. Global temperature
Three frequencies: Milankovitch cycles

12. Fourier Series
General form 𝑓 𝑥 =
1
2
𝑎0 + 𝑎1 cos 𝑥 + 𝑎2 cos 2𝑥 + 𝑎3 cos 3𝑥 + ⋯
+ 𝑏1 sin 𝑥 + 𝑏2 sin 2𝑥 + 𝑏3 sin 3𝑥 + ⋯
𝑓 𝑥 =
1
2
𝑎0 +
𝑛=1
∞
𝑎𝑛 cos 𝑛𝑥 +
𝑛=1
∞
𝑏𝑛sin(𝑛𝑥)
𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥
𝑎𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 cos 𝑛𝑥 𝑑𝑥
𝑏𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 sin 𝑛𝑥 𝑑𝑥
Coefficients
𝑓 𝑥 =
1
2
𝑎0 +
𝑛=1
∞
𝑎𝑛 cos
𝑛𝜋𝑥
𝑙
+
𝑛=1
∞
𝑏𝑛sin(
𝑛𝜋𝑥
𝑙
)
𝑎0 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 𝑑𝑥
𝑎𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 cos
𝑛𝜋𝑥
𝑙
𝑑𝑥
𝑏𝑛 =
1
𝑙 −𝑙
𝑙
𝑓 𝑥 sin
𝑛𝜋𝑥
𝑙
𝑑𝑥
Coefficients

13. Fourier Series
Example: Square Wave 𝑓 𝑥 =
0 − 𝜋 < 𝑥 < 0
1 0 < 𝑥 < 𝜋
𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥
𝑎𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 cos 𝑛𝑥 𝑑𝑥
𝑏𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 sin 𝑛𝑥 𝑑𝑥
a0 = 1, and for all other (n > 0) an = 0

14. Fourier Series
Example: Square Wave 𝑓 𝑥 =
0 − 𝜋 < 𝑥 < 0
1 0 < 𝑥 < 𝜋
𝑎0 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 𝑑𝑥
𝑎𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 cos 𝑛𝑥 𝑑𝑥
𝑏𝑛 =
1
𝜋 −𝜋
𝜋
𝑓 𝑥 sin 𝑛𝑥 𝑑𝑥

15. 𝑓 𝑥 =
0 − 𝜋 < 𝑥 < 0
1 0 < 𝑥 < 𝜋
𝑓 𝑥 =
1
2
𝑎0 +
𝑛=1
∞
𝑎𝑛 cos 𝑛𝑥 +
𝑛=1
∞
𝑏𝑛sin(𝑛𝑥)
𝑎0 = 1
𝑎𝑛 = 0 (𝑛 > 0)
𝑏𝑛 =
2
𝑛𝜋
𝑓𝑜𝑟 𝑜𝑑𝑑 𝑛 (𝑓𝑜𝑟 𝑒𝑣𝑒𝑛 𝑛, 𝑏𝑛 = 0)
𝑓 𝑥 =
1
2
+
2
𝜋
sin 𝑥
1
+
sin 3𝑥
3
+
sin 5𝑥
5
+ ⋯
Fourier Series
Example: Square Wave