Union of intervals

Example

Example about union of intervals

\displaystyle \bigcup_{n=1}^{\infty}\left[ \frac{n+1}{n}, \frac{n+1}{n}+1 \right]
k=1 :\quad \displaystyle \bigcup_{n=1}^{1}\left[ \frac{n+1}{n}, \frac{n+1}{n}+1 \right]
\displaystyle =\left[ \frac{1+1}{1}, \frac{1+1}{1}+1 \right]=\left[ 2, 3 \right]
k=2 :\quad \displaystyle \bigcup_{n=1}^{2}\left[ \frac{n+1}{n}, \frac{n+1}{n}+1 \right]
\displaystyle=\left[ \frac{1+1}{1}, \frac{1+1}{1}+1 \right] \bigcup \left[ \frac{2+1}{2}, \frac{2+1}{2}+1 \right]=\left[ 2, 3 \right]\cup \left[ \frac{3}{2}, \frac{5}{2} \right]
k=3 :\quad \displaystyle \bigcup_{n=1}^{3}\left[ \frac{n+1}{n}, \frac{n+1}{n}+1 \right]
\displaystyle=\left[ \frac{1+1}{1}, \frac{1+1}{1}+1 \right] \bigcup \left[ \frac{2+1}{2}, \frac{2+1}{2}+1 \right] \bigcup \left[ \frac{3+1}{3}, \frac{3+1}{3}+1 \right]
\displaystyle=\left[ 2, 3 \right]\bigcup \left[ \frac{3}{2}, \frac{5}{2} \right]\bigcup \left[ \frac{4}{3}, \frac{7}{3} \right]
\displaystyle=\left[ \frac{3}{2}, 3 \right]
\displaystyle=\left[ \frac{4}{3}, 3 \right]
\displaystyle \bigcup_{n=1}^{k}\left[ \frac{n+1}{n}, \frac{n+1}{n}+1 \right]

Partial union:

\displaystyle 1
\displaystyle + \,\frac{1}{2}
\displaystyle + \,\frac{1}{4}
\displaystyle + \,\frac{1}{8}
\displaystyle +\, \frac{1}{16}
\displaystyle +\, \frac{1}{32}
\displaystyle + \cdots
\displaystyle =2

😃

\displaystyle 1
\displaystyle +\, \frac{1}{2}
\displaystyle + \,\frac{1}{4}
\displaystyle +\, \frac{1}{8}
\displaystyle +\, \frac{1}{16}
\displaystyle +\, \frac{1}{32}
\displaystyle + \cdots

🤔

\displaystyle \overset{\mathrm{?}}{=} 2
\displaystyle 1
\displaystyle +\, \frac{1}{2}
\displaystyle + \,\frac{1}{4}
\displaystyle +\, \frac{1}{8}
\displaystyle +\, \frac{1}{16}
\displaystyle +\, \frac{1}{32}
\displaystyle + \cdots

🤔

\displaystyle \overset{\mathrm{?}}{=} 2
\displaystyle 1
\displaystyle + \,3
\displaystyle + \,6
\displaystyle + \,5
\displaystyle + \,2
\displaystyle +\,4
\displaystyle=21
\displaystyle 1
\displaystyle + \,3
\displaystyle + \,6
\displaystyle + \,5
\displaystyle + \,2
\displaystyle +\,4
\displaystyle=21
\displaystyle 1
\displaystyle - \,\frac{1}{2}
\displaystyle + \,\frac{1}{3}
\displaystyle - \,\frac{1}{4}
\displaystyle +\, \frac{1}{5}
\displaystyle -\, \frac{1}{6}
\displaystyle + \cdots
\displaystyle =\log(2)

😃

\displaystyle 1
\displaystyle - \,\frac{1}{2}
\displaystyle + \,\frac{1}{3}
\displaystyle - \,\frac{1}{4}
\displaystyle +\, \frac{1}{5}
\displaystyle -\, \frac{1}{6}
\displaystyle \neq \log(2)
\displaystyle + \cdots
\displaystyle 1
\displaystyle - \,\frac{1}{2}
\displaystyle + \,\frac{1}{3}
\displaystyle - \,\frac{1}{4}
\displaystyle +\, \frac{1}{5}
\displaystyle -\, \frac{1}{6}
\displaystyle = \text{anything}
\displaystyle + \cdots
3
2
9
=
3
2
9
3
2
9
=
\log
=
3
2
9
4
3
64
=
3
64
4
4
3
64
=
\log
=
4
3
64
b
y
x
=
y
x
b
=
\log
=
b
y
x
y
x
b
y
x
b
=
y
x
2
y
x
2
=
\log
=
2
y
x
y
x
2
y
x
2
\displaystyle \frac{3}{8}+\frac{1}{2}\cos \left(2x\right)+\frac{1}{8}\cos \left(4x\right)

Fourier Series

\cos^4(x)
2
4
1
5
0
3
3
8
2
x
x
x
y
y
y
z
z
z
+
+
+
+
+
+
=
=
=
-3
0
2
2
4
1
5
0
3
3
8
2
x
-3
0
2
2
4
1
5
0
3
3
8
2
x
y
z
-3
0
2
2
4
1
5
0
3
3
8
2
x
x
x
y
y
y
z
z
z
+
+
+
+
+
+
=
=
=
-3
0
2
2
4
1
5
0
3
3
8
2
x
-3
0
2
\left[\right.
\left.\right]
\left[\right.
\left.\right]
=
\left[\right.
\left.\right]
\Rightarrow

💧

😄

💧

😄

\(\log\)

💧

😄

\(\log\)

🔒

🔑

\(\log\)

🔒

🔑

🔒

🔑

\(\log\)

\(\log\)

\(+\)

☁️

💦

☁️

💦

\(\log\)

☁️

💦

\(\log\)

\(-\)

\(\log\)

Union of intervals: Example

By Juan Carlos Ponce Campuzano

Union of intervals: Example

Union of intervals: Example

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