Math 166 - Study Guide: Exam #3
1. L’Hopital’s Rule and Indeterminate Forms
• MUST be of the form
0
0
or
∞
∞
to use L’Hopital’s Rule.
• If you have ∞ − ∞ or 0 × ∞ then use algebra to get into the form above.
• If you have the exponential indeterminate forms (00 , ∞0 , or 1∞ ), use logarithms to get rid of the
exponents. Find the limit of ln(y), then the desired limit is elim ln(y) .
2. Improper Integrals
• Infinite Endpoint: Consider the limit as endpoint approached ∞ or − ∞.
• Unbounded Discontinuity: Consider the limit as the endpoint approached the discontinuity.
• Unbounded Discontinuity between Endpoints: Break up the integral into two integrals
using the discontinuity as an endpoint for both integrals.
3. Sequences
∞
• Sequences are discrete ordered listings of numbers following a specific pattern, {an }n=1 where an
yields a value for each integer plugged in for n.
• Sequences converge if lim an is a finite number. The sequence converges to this limit.
n→∞
• NOTE: The above limit only needs to converge for a sequence to converge. The limit DOES NOT
have to converge to zero in order to have convergence of the overall sequence.
• Squeeze Theorem: The idea is to trap an between a lower and upper bound. If the two limits
of the upper and lower bound converge to the same value, then the limit an converges to the same
value.
sin(n)
∞
.
• Example: Let {an }n=1 , where an =
n
1
sin(n)
1
Clearly, −1 ≤ sin(n) ≤ 1 ⇒ − ≤
≤ .
n
n
n
Then, 0 = lim
n→∞
Thus, lim
n→∞
−1
sin(n)
1
≤ lim
≤ lim
= 0.
n→∞
n→∞ n
n
n
sin(n)
= 0.
n
4. Series
• Similar to sequences above, but the terms are summed together:
∞
X
ak .
k=1
• Nth term test: If lim ak 6= 0, then the series diverges.
k→∞
• Collapsing Sum: Look at the n
th
partial sum Sn =
n
X
ak .
k=1
NOTE: Most of the terms should cancel each other out in the previous step.
n
∞
X
X
Then calculate the limit lim Sn = lim
ak =
ak in order to evaluate the series.
n→∞
n→∞
k=1
1
k=1
• Geometric Series: Consider
∞
X
ark .
k=1
If |r| < 1, then the series converges and
∞
X
k=1
ark =
a
.
1−r
If |r| ≥ 1, then the series diverges.
5. Other Notes:
• I will expect that you know the integrals/derivatives of the basic (polynomial, rational, logarithmic, exponential, roots, trig, and inverse trig) functions.
• As always, it is very important that you READ THE DIRECTIONS for every question.
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