1. Section 3 – Infinite Sequences and Series
Chapter 12
Section 3 – Infinite Sequences and Series

2. Limits
Sequences and Series (like we saw in the last two sections) can be infinite and never end.
However, they can also be approaching a certain number.
When a sequence or series gets closer and closer to a specific number, this is what is called the limit of the sequence or series.

3. Example 1
Estimate the limit of 9/5, 16/4, 65/27, ..., (7n2 + 2)/(2n2 + 3n), ...
Calculators

4. Theorems for Limits Limit of a Sum:
lim 𝑛→∞ 𝑎 𝑛 + 𝑏 𝑛 = lim 𝑛→∞ 𝑎 𝑛 + lim 𝑛→∞ 𝑏 𝑛
Limit of a Difference:
lim 𝑛→∞ 𝑎 𝑛 − 𝑏 𝑛 = lim 𝑛→∞ 𝑎 𝑛 − lim 𝑛→∞ 𝑏 𝑛
Limit of a Product:
lim 𝑛→∞ 𝑎 𝑛 ∙ 𝑏 𝑛 = lim 𝑛→∞ 𝑎 𝑛 ∙ lim 𝑛→∞ 𝑏 𝑛

5. Theorems for Limits Continued
Limit of a Quotient:
lim 𝑛→∞ 𝑎 𝑛 𝑏 𝑛 = lim 𝑛→∞ 𝑎 𝑛 lim 𝑛→∞ 𝑏 𝑛
Limit of a Constant:
lim 𝑛→∞ 𝑐 𝑛 = 𝑐

6. Key Points for determining Limits
If the largest exponents are the same in numerator and denominator, limit = the ratio of the coefficients of the terms with the largest exponent.
If the largest exponent is in the numerator, there is NO limit.
If the largest exponent is in the denominator, the limit = 0

7. Key Points for determining Limits
If a GEOMETRIC (not arithmetic…) sequence has a common ration |r|>1 then its infinite series has no sum, but if |r|<1 then the infinite series has a SUM:
S = 𝒂 𝟏 𝟏−𝒓 for |r| < 1

8. Examples Find each limit in the following examples:
EX 2: limn->∞(1+3n2)/n2
EX 3: limn->∞(5n2+ n-4)/(n2 + 1)
EX 4: limn->∞(4n2+5n+2)/2n

9. Repeating Decimals
All repeating decimals (.6666) can be rewritten as fractions. We can do this because of a formula for infinite series:
S = (a1)/(1-r)
where a1 is the first term of the series and r is the unit fraction similar to the first term

10. Examples EX 5: Write .6666 as a fraction

11. Assignment
Chapter 12, Section 3
pgs #14-30E,43,45