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8.1. SEQUENCES
whose domain is the set of positive integers. In this context, the sequence s 1 , s 2 , s 3 , . . .
would correspond to the function f satisfying f (n) = s n for each positive integer n. This
alternative view will be be useful in many situations.
We will often write the sequence
s 1 , s 2 , s 3 , . . .
using the shorthand notation {s n }. The value s n (alternatively s(n)) is called the nth term
in the sequence. If the terms are all 0 after some fixed value of n, we say the sequence
is finite. Otherwise the sequence is infinite. We will work with both finite and infinite
sequences, but focus more on the infinite sequences. With infinite sequences, we are often
interested in their end behavior and the idea of convergent sequences.
Activity 8.1.
(a) Let s n be the nth term in the sequence 1, 2, 3, . . ..
Find a formula for s n and use appropriate technological tools to draw a graph
of entries in this sequence by plotting points of the form (n, s n ) for some values
of n. Most graphing calculators can plot sequences; directions follow for the
TI-84.
• In the MODE menu, highlight SEQ in the FUNC line and press ENTER.
• In the Y= menu, you will now see lines to enter sequences. Enter a value
for nMin (where the sequence starts), a function for u(n) (the nth term in
the sequence), and the value of u nMin .
• Set your window coordinates (this involves choosing limits for n as well as
the window coordinates XMin, XMax, YMin, and YMax.
• The GRAPH key will draw a plot of your sequence.
Using your knowledge of limits of continuous functions as x → ∞, decide if
this sequence {s n } has a limit as n → ∞. Explain your reasoning.
(b) Let s n be the nth term in the sequence 1,
1
2 ,
1
3 , . . .. Find a formula for s n . Draw
a graph of some points in this sequence. Using your knowledge of limits of
continuous functions as x → ∞, decide if this sequence {s n } has a limit as
n → ∞. Explain your reasoning.
(c) Let s n be the nth term in the sequence 2,
3
2 ,
4
3 ,
5
4 , . . .. Find a formula for s n .
Using your knowledge of limits of continuous functions as x → ∞, decide if
this sequence {s n } has a limit as n → ∞. Explain your reasoning.
⊳
Next we formalize the ideas from Activity 8.1.
8.1. SEQUENCES
whose domain is the set of positive integers. In this context, the sequence s 1 , s 2 , s 3 , . . .
would correspond to the function f satisfying f (n) = s n for each positive integer n. This
alternative view will be be useful in many situations.
We will often write the sequence
s 1 , s 2 , s 3 , . . .
using the shorthand notation {s n }. The value s n (alternatively s(n)) is called the nth term
in the sequence. If the terms are all 0 after some fixed value of n, we say the sequence
is finite. Otherwise the sequence is infinite. We will work with both finite and infinite
sequences, but focus more on the infinite sequences. With infinite sequences, we are often
interested in their end behavior and the idea of convergent sequences.
Activity 8.1.
(a) Let s n be the nth term in the sequence 1, 2, 3, . . ..
Find a formula for s n and use appropriate technological tools to draw a graph
of entries in this sequence by plotting points of the form (n, s n ) for some values
of n. Most graphing calculators can plot sequences; directions follow for the
TI-84.
• In the MODE menu, highlight SEQ in the FUNC line and press ENTER.
• In the Y= menu, you will now see lines to enter sequences. Enter a value
for nMin (where the sequence starts), a function for u(n) (the nth term in
the sequence), and the value of u nMin .
• Set your window coordinates (this involves choosing limits for n as well as
the window coordinates XMin, XMax, YMin, and YMax.
• The GRAPH key will draw a plot of your sequence.
Using your knowledge of limits of continuous functions as x → ∞, decide if
this sequence {s n } has a limit as n → ∞. Explain your reasoning.
(b) Let s n be the nth term in the sequence 1,
1
2 ,
1
3 , . . .. Find a formula for s n . Draw
a graph of some points in this sequence. Using your knowledge of limits of
continuous functions as x → ∞, decide if this sequence {s n } has a limit as
n → ∞. Explain your reasoning.
(c) Let s n be the nth term in the sequence 2,
3
2 ,
4
3 ,
5
4 , . . .. Find a formula for s n .
Using your knowledge of limits of continuous functions as x → ∞, decide if
this sequence {s n } has a limit as n → ∞. Explain your reasoning.
⊳
Next we formalize the ideas from Activity 8.1.