Note 13 Infinite Series - 220629 - 105419
Note 13 Infinite Series - 220629 - 105419
Note 13 Infinite Series - 220629 - 105419
INFINITE SERIES
5.1 SEQUENCES
Sequence is defined by unending succession of numbers, called terms, in a certain order. A sequence
can be written as 𝑎1 , 𝑎2 , 𝑎3 , ⋯ , 𝑎𝑛 .
1, 2, 3, 4, ⋯
1 1 1
, , ,⋯
2 4 8
1, −1, 1, −1, 1, −1, ⋯
𝑎1 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚, 𝑎2 = 𝑠𝑒𝑐𝑜𝑛𝑑 𝑡𝑒𝑟𝑚𝑠 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛.
Definition:
A sequence is a function whose domain is a set of positive integers, denotes as {𝑎𝑛 }∞
𝑛=1 or simply
written as {𝑎𝑛 }.
1 2 3 4 𝑛 ∞
For example: , , , , ⋯ = { }
2 3 4 5 𝑛+1 𝑛−1
1 1 1
b. 1, 3 , 9 , 27 , ⋯
Solution:
5.1.2 Convergence of infinite sequence
Note:
i) If lim 𝑎𝑛 exist then {𝑎𝑛 } converges.
𝑛→∞
ii) If lim 𝑎𝑛 does not exist then {𝑎𝑛 } diverges.
𝑛→∞
Solution:
∞
𝑛2 +1
b. { }
𝑛 𝑛=1
Solution:
𝑎1 + 𝑎2 + 𝑎3 + ⋯ = ∑ 𝑎𝑛
𝑛=1
Note:
i. A sequence is a succession: 𝑎1 , 𝑎2 , 𝑎3 , ⋯ , 𝑎𝑛
𝑛 ∞ 1 2 3 4
{𝑎𝑛 } = { } = , , , ,⋯
2𝑛 + 1 𝑛 = 1 3 5 7 9
ii. A series is a sum of a succession: 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛 + ⋯
∞ ∞
𝑛 1 2 3 4
∑ 𝑎𝑛 = ∑ = + + + + ⋯ = 𝑠𝑛
2𝑛 + 1 3 5 7 9
𝑛=1 𝑛=1
𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + 𝑎𝑟 3 + ⋯ 𝑎𝑟 𝑛−1 , 𝑎 ≠ 0
where each term is obtained by multiplying the preceding one by a constant r, called ratio. Geometric
series can simply be written as ∑∞ 𝑛
𝑛=1 𝑟 . For example:
2, 4, 8, 16, ⋯ 𝑎 = 2, 𝑟 = 2
4 4 4 4 4 1
, , , ,⋯ 𝑎 = ,𝑟 =
3 32 33 34 3 3
∞
1 𝑛+1 1 1
∑( ) 𝑎 = ,𝑟 =
3 9 3
1
∑∞ 𝑛
𝑛=1 𝑟 converges to S if |𝑟| < 1 and diverges if |𝑟| ≥ 1
𝑎
Sum 𝑆 = 1−𝑟,where a is the first term, r is the ratio
Example 3: Determine if the following series converges or diverges. If it converges, find its sum.
3 3𝑛
a. ∑∞
1 (2)
Solution:
22𝑛+1
b. ∑∞
1 3𝑛
Solution:
2
c. ∑∞
1 5𝑛
Solution:
i. ∑∞ ∞
𝑛=1 𝑐𝑎𝑛 = 𝑐 ∑𝑛=1 𝑎𝑛 = 𝑐𝑆1 ii. ∑∞ ∞ ∞
𝑛=1 𝑎𝑛 + 𝑏𝑛 = ∑𝑛=1 𝑎𝑛 + ∑𝑛=1 𝑏𝑛
= 𝑆1 + 𝑆2
Let ∑∞ ∞ ∞
𝑛=1 𝑎𝑛 converges and ∑𝑛=1 𝑏𝑛 diverges. Then ∑𝑛=1 𝑎𝑛 + 𝑏𝑛 diverges.
Example 5: determine if the following series converges or diverges. If it converges, find its sum.
1 𝑛+1 1 3𝑛
a. ∑∞
1 (3) + (3)
Solution:
𝑒 𝑛
b. 0.212121 ⋯ + ∑∞
1 ( ) 3
Solution:
5.2.2 Telescoping series
A telescoping series is a series where the internal terms cancel each other. Usually a telescoping series
is in the form ∑∞
1 𝑎𝑛 − 𝑏𝑛 .
Example 6: Determine if the following series converges or diverges. If it converges, find its sum.
1 1
a. ∑∞
𝑛=1 −
4𝑛 4 𝑛+1
Solution:
1
b. ∑∞
𝑛=4 𝑛2 −5𝑛+6
Solution:
5.2.3 Harmonic series
1
Definition: A harmonic series ∑∞
1 𝑛 is a diverging series.
If the series ∑∞
𝑛=1 𝑎𝑛 converges, then lim 𝑎𝑛 = 0
𝑛→∞
For example:
1 1 1 1
∑∞
𝑛=1 𝑛 = 1 + 2 + 3 + 4 + ⋯ is divergent series
Solution:
1
b. ∑∞
𝑛=4 𝑛
Solution:
5.2.4 P-series
1 1 1 1
A p-series or hyper-harmonic series is an infinite series of the form 1 + 2𝑝 + 3𝑝 + 4𝑝 + ⋯ = ∑∞
1 𝑛𝑝
1
∑∞
𝑛=1 converges, if 𝑝 > 1 and diverges if 0 < 𝑝 ≤ 1
𝑛𝑝
Example 8: Determine if the series is convergent or divergent.
𝑛2
a. ∑∞
𝑛=1 𝑛4
Solution:
𝑠𝑒𝑐 2 𝑛−𝑡𝑎𝑛2 𝑛
b. ∑∞
𝑛=1 𝑛
Solution:
cos(2𝜋𝑛)
c. ∑∞
𝑛=1 𝑛3
Solution:
1
d. If 𝑝 = 1, ∑∞
𝑘=1 𝐾𝑝
Solution: