Calculus 2-Sequences, guaranteed 100% Pass

1


Sequences
A sequence is a list of numbers:

𝑎1, 𝑎2, 𝑎3, 𝑎4, … , 𝑎𝑛 , …
This sequence may be denoted by {𝑎1, 𝑎2, 𝑎3, 𝑎4, … , 𝑎𝑛 , … }, {𝑎𝑛 }, or just 𝑎𝑛 .


Ex.
1 1 1 1
a. 1, , , , … , , … where 𝑎𝑛 = 1/𝑛 ; or {1/𝑛}
2 3 4 𝑛


1 1 1 1 1
b. 1, − , , − , ,… in this sequence 𝑎𝑛 = (−1)𝑛−1 ( ); or
2 4 8 16 2𝑛−1
1
{𝑎𝑛 } = {(−1)𝑛−1 ( 𝑛−1 )}
2



c. √5, √6, √7, √8, … in this sequence 𝑎𝑛 = √𝑛 + 4; or
{𝑎𝑛 } = {√𝑛 + 4}


Ex. Find a formula for the 𝑛th term of:
2 3 4 5 6 7
a. {− , , − , , − , , … }
3 4 5 6 7 8
b. {−2, 3, −2, 3, −2, 3, … }



𝑛+1
a. 𝑎𝑛 = (−1)𝑛
𝑛+2
b. 𝑎2𝑛−1 = −2
𝑎2𝑛 = 3

, 2


Some sequences don’t have an easy formula for the 𝑛th term:
Ex. The Fibonacci sequence:

𝑎1 = 1, 𝑎2 = 1, 𝑎3 = 2, 𝑎4 = 3, 𝑎5 = 5, 𝑎6 = 8, 𝑎7 = 13,
𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2


Given a sequence of numbers {𝑎𝑛 }, we can ask if the sequence converges to
some number.

Def. A sequence {𝑎𝑛 } has a limit 𝑳, written lim 𝑎𝑛 = 𝐿 or 𝑎𝑛 → 𝐿 as
𝑛→∞
𝑛 → ∞, if for every 𝜖 > 0 there is an integer 𝑁 such that if 𝑛 > 𝑁, then
|𝑎𝑛 − 𝐿| < 𝜖. If lim 𝑎𝑛 = 𝐿 < ∞, then we say the sequence {𝑎𝑛 } converges.
𝑛→∞
If a sequence doesn’t converge, then we say it diverges.




𝐿+𝜖
𝐿
𝐿−𝜖



𝑁




1 1 1 1 1 1 1
Ex. { } = 1, , , , , ,… converges to 0 since lim = 0.
𝑛 2 3 4 5 6 𝑛→∞ 𝑛