Pre-Calculus
Sequences and Series Notes
-Arithmetic-
1.
Find the next four terms in the arithmetic sequence -5, -2, 1, …
2.
Find the 47th term in the arithmetic sequences -4, -1, 2, 5, …
3.
Find the first term in the arithmetic sequence for which 𝑎
19
= 42 𝑎𝑛𝑑 𝑑 = −
2
3
4.
Write an arithmetic sequence that has five arithmetic means between 4.9 and 2.5
5.
Find the sum of the first 60 terms of the arithmetic series 9 + 14 + 19 + … + 304

-Geometric-
1.
Determine the common ratio and find the next three terms in the sequence 1, −
1
2
,
1
4
, …
2.
Determine the next three terms in each sequence 𝑟 − 1, − 3𝑟 + 3, 9𝑟 − 9, …
3.
Find an approximation for the 23rd term in the sequence 256, -179.2, 125.44, …
4.
Write a sequence that has two geometric means between 48 and -750
5.
Find the sum of the first ten terms of the geometric series 16 – 48 + 144 – 432 +…

-Infinite and Limits-
1.
Find each limit: lim 𝑛→∞
1+3𝑛
2 𝑛 2
2.
Find each limit: lim 𝑛→∞
5𝑛
2 𝑛 2
+𝑛−4
+1
3.
Find each limit:
lim 𝑛→∞
2+5𝑛+4𝑛
2
2𝑛
4.
Find the sum of the infinite geometric series 21 − 3 +
3
7
− ⋯
5.
Write 0. 762 as a fraction.

Convergent and Divergent Series
•
If an infinite series has a sum, or limit, the series converges.
•
If the series is not convergent, it is divergent
How to determine if the series is convergent or divergent
•
1 st determine if the series is a geometric or arithmetic series.
•
2 nd
Find r if geometric. If |r|<1 it is convergent. If |r|>1 it is divergent.
•
If arithmetic then automatically divergent because it has no limit.
1.
Determine if the series converges or diverges −
1
2
+
1
4
−
1
8
1
+
16
− ⋯
2.
Determine if the series converges or diverges 2 + 4 + 8 + 16 + ⋯
3.
Determine if the series converges or diverges 10 + 8.5 + 7 + 5.5 + ⋯
The Ratio Test:
1 st
: You must find 𝑎 𝑛
and 𝑎 𝑛+1
2 nd : You find r by taking the lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛
3 rd
: You check to see which is true: if r < 1 it is convergent if r > 1 it is divergent if r = 1 it has no information

4.
Determine if the series converges or diverges
1
2
+
2
4
+
3
8
+
4
16
+ ⋯
5.
Determine if the series converges or diverges
1
2
+
2
3
+
3
4
+
4
5
+ ⋯
6.
Determine if the series converges or diverges 1 +
1
2!
+
1
3!
+
1
4!
+ ⋯