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Series


Definition: If {ano} is a sequence, so the infinite sum:

The1 + a2 + a3 +… + Ano +… =

it's called series.

Each number Thei is a term in the series;

Theno is the generic term n.

To define the sum of infinite installments, we consider the partial sums

s1 = a1
s2 = a1 + a2
s3 = a1 + a2 + a3
------------------------
sno = a1 + a2 + a3 +… + An-1 + ano

And the sequence of partial sums

s1, S2, S3,… , Sno,…

If this sequence has limit S, then the series converge and your sum is S.

That is: If , then the series converges and its sum is the1+ a2+ a3+… + Ano… = S

If the sequence {Sno} has no limit, so the series differ.

If the series converge then .

Note: * The reciprocal of this theorem is false, that is, there are series whose generic term tends to zero and which are not convergent.

* It is worth the counter: "if the limit is not zero then the series no converge", which constitutes the following test.

Divergence Test

Given the series ,
diverges.

Next: Geometric Series