Intro to Calculus Limits of Sequences Areas Katherine Mihalczo Precalc Survey, 7th

Limits of Sequences The most important thing when finding the limit of a sequence is having a n a n is the “n th ” term of the sequence a 1, a 2, a 3, a 4, ….. a n By finding the limit of the a n 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

Riemann Sum Left handRight hand OverestimatesUnderestimates Average

Example First we take the right hand sumThen we take the left hand sum 123 Equation: f(x) = x Then we average them to find a more accurate estimation of the area

Example Continued First:

Then: 6.875

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

Equation The sum of the functions Area Limit as n approaches infinity The first rectangle “n” rectangles