Sequences, Series, Limits, Continuity, and Derivatives in Calculus
Sequences
If a sequence fails to approach any single number, then it does not have a limit, or the limit is undefined.
Common Behaviors for a Sequence
- Approach a Limit L
- Jump around without settling anywhere
- Grow toward ± ∞
Ex/ 
Limit is 1
{1,-1,1,-1….} No limit.
(1,2,4,8,16….} Diverges to ∞ …no limit
- Diverging -> gets bigger, approaching infinity
- Converging ->to a limit.
If a series has a limit, we say it converges; otherwise, it diverges.
Common Limits
Series
Harmonic Series
–> Divergent; No limit.
P-Series Test
If p is a real number, then the series:
- Converges for p>1
- Diverges for p≤1
Geometric Series
Always have the form:
Infinite
- d= First term of the Sequence
- r = ratio of the Sequence (second term/ first term)
Finite
To find if a series is geometric: divide the second term by the first, and if that number equals the third term divided by the second, then it is geometric
Rule:
Constants that are:
- Less than minus 1→Jumps around while diverging
- Equal to minus 1 → Jumps around without converging
- Strictly between -1, and 1→ Converges to 0
- Equal to 1→ Stays at 1
- Greater than 1 → Grows infinitely
Infinite Geometric Series Formula
Bouncing Ball Problem
Suppose a ball is dropped 3 meters and bounces half as high every time. Find the total vertical distance traveled.
3m 1.5 0.75 0.375
3+1.5+0.75+0.375…….
Can be rewritten as:
3+3(1/2)+3(1/4)+3(1/8)…..
Can be rewritten as:
Can be rewritten as:
Solve using geometric series formula
Total Vertical Distance Down
Vertical distance traveled upwards is the down distance minus the initial drop
6-3 = 3
Add Down and Up distance
6+3=9 The total distance traveled.
Applications To Finance
Formula for Compound Interest with Deposits
- First deposit made at the end of the first compounding period
i = Interest/ Compounding periods in a year
c= Total compounding periods
- First deposit made at the beginning of the first compounding period
Formula for the Amount of a Loan Payment
P= principal p= payment
m= multiplier ( interest/compounds in a year/100 +1)
c= number of compounding periods
Formula to find number of payments
Formula to find the original amount of the loan
Minimum amount of a Payment
If we make payments p on a loan P with multiplier m, then p>P(m-1) for the loan to ever be repaid
If i% is the interest rate in each period p>p(i/100) for the loan to ever be repaid
Limits and Continuity
Heavyside Function
Look at the y value to find the limit
If the left limit equals the right limit, then the limit exists; if not, it doesn’t exist.
Note:
For Limits Log(x)
Polynomials are continuous, so to find the limit, plug the number in x is approaching
Rational fractions:
- If the top is growing faster → limit is ∞
- If the bottom is growing faster → limit is 0
Continuity
- A function has to have a limit
- Left side and right side limits must equal
- Must be defined at all points (≥)
Polynomial Functions are continuous everywhere
ex/ Find all values of ‘a’ that make the function continuous
Derivatives
ex/ Use the definition of a derivative to f'(x)
Derivative Rules
Power Rule: 
Product Rule
Quotient Rule
** Leave the bottom unsimplified
ln(x)’ = 1/x logb(x)’ = 1/ln(b)•x
e^x’ = e^x c^x’ = ln(c)• c^x
Reciprocal Rule: 
Chain Rule:
