Intro to Calculus

Limits of Sequences

Areas

Katherine MihalczoPrecalc Survey, 7th

Limits of Sequences

The most important thing when finding the limit of a sequence is

having an

an is the “nth” term of the sequence

a1, a2, a3, a4, ….. an

By finding the limit of the an term, we find the limit of the sequence!

Example

Find the limit of

= 0

Archimedes

Areas

By dividing shapes into rectangles we can approximate the area

Method of exhaustion:

keep making the areas of the rectangles smaller until satisfied that the result is accurate enough

Like the top of a barrel

A circle made from rectangular pieces of wood

Rectangles and Archimedes

Pro:

we know the approximate area

Con:

not completely accurate

exhausting…

Solution 1

Smaller rectangles

Still the problem of accuracy….

Solution 2

Riemann Sum More accurate

Area Under a Curve

hxAhxAxf )()()(

Riemann Sum)(

n

kaf

n

ahbArect

Left handRight hand

Overestimates Underestimates

Average

Example

First we take the right hand sum Then we take the left hand sum

1 2 3

Equation: f(x) = x 2

1 2 3

Then we average them to find a more accurate estimation of the area

Example Continued

First:

11.375

Then:

6.875

Lastly, we take the average of the two sums to find the area:

Equation

xfAn

kkn x

)(lim1

n

abx

The sum of the functions

Area

Limit as n approaches infinity

The first rectangle

“n” rectangles