Foundations of Data Science

f(n) is O(n) but in bounding the lower order term, we write f(n) = 2n + o(n). Finally,
f(n)
f(n)
we write f(n) ∼ g(n) if lim = 1 and say f(n) is ω(g(n)) if lim = ∞. The
n→∞ g(n)
n→∞ g(n)
difference between f(n) being Θ(g(n)) and f(n) ∼ g(n) is that in the first case f(n) and
g(n) may differ by a multiplicative constant factor.
12.2 Useful relations
Summations
n∑
a i = 1 + a + a 2 + · · · = 1 − an+1
1 − a , a ≠ 1
∞∑
a i = 1 + a + a 2 + · · · = 1
1 − a , |a| < 1
∞∑
ia i = a + 2a 2 + 3a 3 a · · · =
(1 − a) , |a| < 1
2
∞∑
i 2 a i = a + 4a 2 + 9a 3 a(1 + a)
· · · =
(1 − a) , |a| < 1
3
n∑ n(n + 1)
i =
2
n∑
i 2 n(n + 1)(2n + 1)
=
6
∞∑ 1
i = π2
2 6
i=0
i=0
i=0
i=0
i=1
i=1
i=1
We prove one equality.
∑
Write S = ∞ ia i .
Thus,
i=0
∞∑
ia i = a + 2a 2 + 3a 3 · · · =
i=0
S − aS =
aS =
∞∑
ia i+1 =
i=0
∞∑
ia i −
i=1
a
, provided |a| < 1.
(1 − a)
2
∞∑
(i − 1)a i .
i=1
∞∑
(i − 1)a i =
i=1
∞∑
a i =
i=1
a
1 − a ,
from which the equality follows. The sum ∑ i 2 a i can also be done by an extension of this
i
method (left to the reader). Using generating functions, we will see another proof of both
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