This document defines and explains several types of infinite series and tests used to determine if they converge or diverge. It introduces infinite series and sequences of partial sums. It then discusses tests for convergence/divergence including the geometric series test, limit comparison test, ratio test, root test, alternating series test, absolute convergence test, and radius of convergence for power series. Types of series covered include Taylor series and Maclaurin series expansions.

1. Infinite Series
∑ 𝑢1 + 𝑢2 + 𝑢3 + ⋯ … … …
∞
𝑛=1
Sequence of Partial Sums
𝑆1 = 𝑢1
𝑆2 = 𝑢1 + 𝑢2
𝑆3 = 𝑢1 + 𝑢2 + 𝑢3
……………..
𝑆 𝑛 = 𝑢1 + 𝑢2 + ⋯ + 𝑢 𝑛 etc.
1 if the sequence of partial sums (𝑠 𝑛) converges
then the series ∑ 𝑢 𝑛
∞
𝑛=1 converges
2 lim 𝑛→∞ 𝑆 𝑛 is the sum of the series
Geometric Series
∑ 𝑎𝑟 𝑛
∞
𝑛=1
= 𝑎 + 𝑎𝑟 + 𝑎𝑟2
+ ⋯ , (𝑎 ≠ 0)
1 Converges if |𝑟| < 1
2 Diverges if |𝑟| ≥ 1
Necessary condition for convergence
𝐴𝑛 𝑖𝑛𝑓𝑖𝑛𝑡𝑒 𝑠𝑒𝑟𝑖𝑒𝑠
∑ 𝑢 𝑛
∞
𝑛=1
𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒔 𝑡ℎ𝑒𝑛 lim
𝑛→∞
𝑢 𝑛 = 0
𝑖𝑓 lim
𝑛→∞
𝑢 𝑛 ≠ 0 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑠𝑒𝑟𝑖𝑒𝑠 𝑫𝒊𝒗𝒆𝒓𝒈𝒆𝒔
P – Test
lim
𝑛→∞
1
𝑛 𝑝
𝑖𝑠 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡 𝑖𝑓 𝑝 > 1, 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠 𝑖𝑓 𝑝 ≤ 1
Comparison Test
If ∑ 𝑢 𝑛 and If ∑ 𝑣 𝑛 are two positive term series
then if 𝑢 𝑛 ≤ 𝑣 𝑛 ∀ 𝑛 ≥ 𝑚, 𝑚 ∈ 𝑁 then
a) If ∑ 𝑣 𝑛 is convergent then ∑ 𝑢 𝑛 is also
convergent
b) If ∑ 𝑢 𝑛 is divergent then ∑ 𝑣 𝑛 is also divergent
Limit form of comparison test:
lim 𝑛→∞
𝑢 𝑛
𝑣 𝑛
= 𝑙 ≠ 0 is a non-zero number.
Then ∑ 𝑢 𝑛 & ∑ 𝑣 𝑛 converges or diverges together.
D’Alembert’s Ratio Test
If ∑ 𝑢 𝑛 is a positive term series such that
lim 𝑛→∞
𝑢 𝑛+1
𝑢 𝑛
= 𝑙 then the series
a) Converges if 𝑙 < 1
b) Diverges if 𝑙 > 1
c) Test fails if 𝑙 = 1
Root Test
If ∑ 𝑢 𝑛 is a positive term series such that
lim 𝑛→∞(𝑢 𝑛)
1
𝑛 = 𝑙 then the series
a) Converges if 𝑙 < 1
b) Diverges if 𝑙 > 1
c) Test fails if 𝑙 = 1
Alternating series
A series whose terms are alternatively positive
and negative
Leibnitz Test
If the alternating series 𝑢1 − 𝑢2 + 𝑢3 − ⋯ is such
that
a) 𝑢 𝑛+1 ≤ 𝑢 𝑛, ∀𝑛 and
b) lim 𝑛→∞ 𝑢 𝑛 = 0
Then the series converges
Absolute convergence
A series ∑ 𝑢 𝑛
∞
𝑛=1 is called absolutely
convergent if ∑ |𝑢 𝑛|∞
𝑛=1 is convergent.
If ∑ 𝑢 𝑛
∞
𝑛=1 is convergent and ∑ |𝑢 𝑛|∞
𝑛=1 is
divergent we call the series conditionally convergent.
Taylor’s Series
The Taylor’s Series expansion of 𝑓(𝑥) about
𝑥 = 𝑎 is
𝑓(𝑥) = 𝑓(𝑎) +
𝑥 − 𝑎
1!
𝑓′(𝑎) +
(𝑥 − 𝑎)2
2!
𝑓′′(𝑎)
+
(𝑥 − 𝑎)3
3!
𝑓′′′(𝑎) + ⋯ … ..
Maclaurin’s Series
Put 𝑎 = 0 in Taylor’s series
𝑓(𝑥) = 𝑓(0) + 𝑥𝑓′(0) +
𝑥2
2!
𝑓"(0) +
𝑥3
3!
𝑓′′′(0) + ⋯.
Radius of Convergence
If ∑ 𝑎 𝑛 𝑥 𝑛∞
𝑛=1 is a power series which
converges absolutely for |𝑥| < 𝑅 and those absolute
term diverges for |𝑥| > 𝑅 and |𝑥| = 𝑅 if either
converges or diverges absolutely. Then R is knowns
as the Radius of convergence of the power series.