THE SEQUENCESIntroduction

Introduction• A sequence is a particular function which has in input only natural

numbers and in output real numbers. Is used instead of and the notationisinstead of .

Like functions, we can draw the graph of a sequence in Cartesian coordinate system. For example:

Definitions:• A sequenceissaidlowerboundedwhenthereis a

realnumberthatislessthananyvalue of the sequence.

Note that in this case

• A sequenceissaidupperboundedwhenthereis a realnumberthatisalwaysgraterthananyvalue of the sequence.

Note that in this case

• A sequence is called bounded when the sequence is simultaneously upper an lower bounded.

Note that in this case

A sequence is called:• Monotonic increasing if each term of the sequence is grater than or

equal to the previous one• Monotonic decreasing if each term of the sequence is less than or equal

to the previous one

Monotonic increasing Monotonic decreasing

LIMIT OF SEQUENCESHow is the overall graphic of a sequence?

In studying a sequence we may be interested in what happens to the terms as we increase more and more the

value.

• Convergence means that the terms keep getting closer and closer to a particular number.

• Divergence means that the terms keep getting bigger towards infinity, or smaller towards negative infinity.

• Indeterminate means that the terms don’t converge neither diverge.

Convergence definition• When becomes bigger and bigger, we say that a sequence

converges to a value if for any tiny positive number you can choose, exists a natural number so that are all between and .

Divergence definitionWhen becomes bigger and bigger, we say that a sequence diverges to when for any positive number you choose, exists a natural number so that are all bigger than (divergence toward ) or are all less than (divergence toward )

𝑎𝑛=𝑛3

𝑀=2500

Created by Erasmus+ M.A.T.H.S.

Geogebra team studentsGiovanni Montanari

Simone MattioliJernie Pasahol

I.T.T.S. «A. Volta» Perugia2015/2016