Sequences, Series, and Summation Methods
Your Name
September 7, 2025
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Outline
1 Sequences and Series
2 Convergence and Divergence
3 Cesàro Summation
4 Abel Summation
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Sequences
A sequence is a function from the natural numbers N to the real numbers
R.
Example: an = n1 .
Limit of a sequence:
lim an = L
n→∞
if for every ϵ > 0, there exists N such that n > N =⇒ |an − L| < ϵ.
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Series
A series is the sum of the terms of a sequence:
∞
X
S = a1 + a2 + a3 + · · · = an
n=1
Convergent if S tends to a finite limit.
Divergent if S tends to ∞, −∞, or oscillates.
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Convergent Series
Example:
∞
X 1
=1
2n
n=1
This is a geometric series with ratio |r | < 1.
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Divergent and Oscillatory Series
Divergent: 1 + 2 + 3 + 4 + · · · → ∞.
Oscillatory: 1 − 1 + 1 − 1 + · · · does not settle to a finite value.
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Cesàro Summation
Definition
P
A series un is Cesàro summable if the arithmetic means of its partial
sums converge.
Example: 1 − 1 + 1 − 1 + · · ·
Partial sums: 1, 0, 1, 0, . . . (oscillatory)
Cesàro means: 1, 21 , 23 , 21 , . . . → 12 .
Hence Cesàro sum = 12 .
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Abel Summation
Definition
P
A series un is Abel summable to S if
∞
X
lim un x n = S.
x→1−
n=0
Example: 1 − 1 + 1 − 1 + · · ·
1
Consider f (x) = 1 − x + x 2 − x 3 + · · · = 1+x , for |x| < 1.
As x → 1− , f (x) → 21 .
So Abel sum = 21 .
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Conclusion
Sequences and series form the foundation of real analysis.
Divergent series can often be assigned values using summation
methods.
Cesàro and Abel are two powerful summation techniques.
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